L(s) = 1 | + 12·5-s − 140·7-s + 288·11-s + 300·13-s − 1.02e3·17-s + 1.32e3·23-s − 946·25-s + 1.47e3·31-s − 1.68e3·35-s − 300·37-s − 3.48e3·41-s − 6.36e3·43-s + 4.80e3·47-s + 9.80e3·49-s + 3.90e3·53-s + 3.45e3·55-s − 1.15e4·61-s + 3.60e3·65-s − 920·67-s − 3.60e3·71-s + 2.96e3·73-s − 4.03e4·77-s − 1.45e3·81-s + 1.27e4·83-s − 1.22e4·85-s − 4.20e4·91-s − 1.56e4·97-s + ⋯ |
L(s) = 1 | + 0.479·5-s − 2.85·7-s + 2.38·11-s + 1.77·13-s − 3.52·17-s + 2.49·23-s − 1.51·25-s + 1.53·31-s − 1.37·35-s − 0.219·37-s − 2.07·41-s − 3.43·43-s + 2.17·47-s + 4.08·49-s + 1.38·53-s + 1.14·55-s − 3.10·61-s + 0.852·65-s − 0.204·67-s − 0.714·71-s + 0.555·73-s − 6.80·77-s − 2/9·81-s + 1.84·83-s − 1.69·85-s − 5.07·91-s − 1.65·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.024809971\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024809971\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 5 | \( 1 - 12 T + 218 p T^{2} + 108 p^{3} T^{3} + 2922 p^{3} T^{4} + 108 p^{7} T^{5} + 218 p^{9} T^{6} - 12 p^{12} T^{7} + p^{16} T^{8} \) |
good | 7 | \( 1 + 20 p T + 200 p^{2} T^{2} + 581620 T^{3} + 33605888 T^{4} + 1763570380 T^{5} + 86703063000 T^{6} + 4131656135220 T^{7} + 197285678423038 T^{8} + 4131656135220 p^{4} T^{9} + 86703063000 p^{8} T^{10} + 1763570380 p^{12} T^{11} + 33605888 p^{16} T^{12} + 581620 p^{20} T^{13} + 200 p^{26} T^{14} + 20 p^{29} T^{15} + p^{32} T^{16} \) |
| 11 | \( ( 1 - 144 T + 28190 T^{2} - 3812736 T^{3} + 707867034 T^{4} - 3812736 p^{4} T^{5} + 28190 p^{8} T^{6} - 144 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 13 | \( 1 - 300 T + 45000 T^{2} - 9783300 T^{3} + 1975625584 T^{4} - 89578863900 T^{5} - 14173012665000 T^{6} + 4995937878137100 T^{7} - 1069096419384907554 T^{8} + 4995937878137100 p^{4} T^{9} - 14173012665000 p^{8} T^{10} - 89578863900 p^{12} T^{11} + 1975625584 p^{16} T^{12} - 9783300 p^{20} T^{13} + 45000 p^{24} T^{14} - 300 p^{28} T^{15} + p^{32} T^{16} \) |
| 17 | \( 1 + 60 p T + 1800 p^{2} T^{2} + 237174660 T^{3} + 108369519248 T^{4} + 40362998002140 T^{5} + 12922343722431000 T^{6} + 4220935792803263460 T^{7} + \)\(13\!\cdots\!38\)\( T^{8} + 4220935792803263460 p^{4} T^{9} + 12922343722431000 p^{8} T^{10} + 40362998002140 p^{12} T^{11} + 108369519248 p^{16} T^{12} + 237174660 p^{20} T^{13} + 1800 p^{26} T^{14} + 60 p^{29} T^{15} + p^{32} T^{16} \) |
| 19 | \( 1 - 807568 T^{2} + 310052205148 T^{4} - 3844553186125264 p T^{6} + 31886389812699488470 p^{2} T^{8} - 3844553186125264 p^{9} T^{10} + 310052205148 p^{16} T^{12} - 807568 p^{24} T^{14} + p^{32} T^{16} \) |
| 23 | \( 1 - 1320 T + 871200 T^{2} - 615508680 T^{3} + 418959700676 T^{4} - 196964604374040 T^{5} + 84421054122472800 T^{6} - 40244644108817989560 T^{7} + \)\(19\!\cdots\!66\)\( T^{8} - 40244644108817989560 p^{4} T^{9} + 84421054122472800 p^{8} T^{10} - 196964604374040 p^{12} T^{11} + 418959700676 p^{16} T^{12} - 615508680 p^{20} T^{13} + 871200 p^{24} T^{14} - 1320 p^{28} T^{15} + p^{32} T^{16} \) |
| 29 | \( 1 - 68708 p T^{2} + 3072346485928 T^{4} - 3101195428283475004 T^{6} + \)\(25\!\cdots\!70\)\( T^{8} - 3101195428283475004 p^{8} T^{10} + 3072346485928 p^{16} T^{12} - 68708 p^{25} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 - 736 T + 908620 T^{2} + 48218336 p T^{3} - 1098717769706 T^{4} + 48218336 p^{5} T^{5} + 908620 p^{8} T^{6} - 736 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( 1 + 300 T + 45000 T^{2} + 3714255300 T^{3} + 1698846223216 T^{4} + 1157016354644700 T^{5} + 7168503043137735000 T^{6} + \)\(86\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!46\)\( T^{8} + \)\(86\!\cdots\!00\)\( p^{4} T^{9} + 7168503043137735000 p^{8} T^{10} + 1157016354644700 p^{12} T^{11} + 1698846223216 p^{16} T^{12} + 3714255300 p^{20} T^{13} + 45000 p^{24} T^{14} + 300 p^{28} T^{15} + p^{32} T^{16} \) |
| 41 | \( ( 1 + 1740 T + 2168144 T^{2} - 1258319580 T^{3} - 6938248543074 T^{4} - 1258319580 p^{4} T^{5} + 2168144 p^{8} T^{6} + 1740 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 43 | \( 1 + 6360 T + 20224800 T^{2} + 50414754840 T^{3} + 88723310157796 T^{4} + 61909663637634120 T^{5} - \)\(12\!\cdots\!00\)\( T^{6} - \)\(66\!\cdots\!20\)\( T^{7} - \)\(16\!\cdots\!94\)\( T^{8} - \)\(66\!\cdots\!20\)\( p^{4} T^{9} - \)\(12\!\cdots\!00\)\( p^{8} T^{10} + 61909663637634120 p^{12} T^{11} + 88723310157796 p^{16} T^{12} + 50414754840 p^{20} T^{13} + 20224800 p^{24} T^{14} + 6360 p^{28} T^{15} + p^{32} T^{16} \) |
| 47 | \( 1 - 4800 T + 11520000 T^{2} - 36388257600 T^{3} + 122277134229028 T^{4} - 247989407219755200 T^{5} + \)\(44\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!38\)\( T^{8} - \)\(10\!\cdots\!00\)\( p^{4} T^{9} + \)\(44\!\cdots\!00\)\( p^{8} T^{10} - 247989407219755200 p^{12} T^{11} + 122277134229028 p^{16} T^{12} - 36388257600 p^{20} T^{13} + 11520000 p^{24} T^{14} - 4800 p^{28} T^{15} + p^{32} T^{16} \) |
| 53 | \( 1 - 3900 T + 7605000 T^{2} + 19637082300 T^{3} - 38989452710992 T^{4} - 168962964131814300 T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - \)\(67\!\cdots\!00\)\( T^{7} - \)\(65\!\cdots\!42\)\( T^{8} - \)\(67\!\cdots\!00\)\( p^{4} T^{9} + \)\(11\!\cdots\!00\)\( p^{8} T^{10} - 168962964131814300 p^{12} T^{11} - 38989452710992 p^{16} T^{12} + 19637082300 p^{20} T^{13} + 7605000 p^{24} T^{14} - 3900 p^{28} T^{15} + p^{32} T^{16} \) |
| 59 | \( 1 - 62483932 T^{2} + 1869161451706168 T^{4} - \)\(36\!\cdots\!64\)\( T^{6} + \)\(50\!\cdots\!70\)\( T^{8} - \)\(36\!\cdots\!64\)\( p^{8} T^{10} + 1869161451706168 p^{16} T^{12} - 62483932 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 + 5772 T + 32102608 T^{2} + 78794192484 T^{3} + 342771003109470 T^{4} + 78794192484 p^{4} T^{5} + 32102608 p^{8} T^{6} + 5772 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( 1 + 920 T + 423200 T^{2} + 118229081560 T^{3} + 171493895006948 T^{4} - 2931027001349315960 T^{5} + \)\(42\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} - \)\(43\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!40\)\( p^{4} T^{9} + \)\(42\!\cdots\!00\)\( p^{8} T^{10} - 2931027001349315960 p^{12} T^{11} + 171493895006948 p^{16} T^{12} + 118229081560 p^{20} T^{13} + 423200 p^{24} T^{14} + 920 p^{28} T^{15} + p^{32} T^{16} \) |
| 71 | \( ( 1 + 1800 T + 43161524 T^{2} - 138273786600 T^{3} + 600779517032166 T^{4} - 138273786600 p^{4} T^{5} + 43161524 p^{8} T^{6} + 1800 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 73 | \( 1 - 2960 T + 4380800 T^{2} + 60676347920 T^{3} + 565797858385148 T^{4} - 6766451224989568720 T^{5} + \)\(19\!\cdots\!00\)\( T^{6} - \)\(72\!\cdots\!80\)\( T^{7} - \)\(46\!\cdots\!02\)\( T^{8} - \)\(72\!\cdots\!80\)\( p^{4} T^{9} + \)\(19\!\cdots\!00\)\( p^{8} T^{10} - 6766451224989568720 p^{12} T^{11} + 565797858385148 p^{16} T^{12} + 60676347920 p^{20} T^{13} + 4380800 p^{24} T^{14} - 2960 p^{28} T^{15} + p^{32} T^{16} \) |
| 79 | \( 1 - 155156984 T^{2} + 13875289326796540 T^{4} - \)\(84\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!94\)\( T^{8} - \)\(84\!\cdots\!16\)\( p^{8} T^{10} + 13875289326796540 p^{16} T^{12} - 155156984 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( 1 - 12720 T + 80899200 T^{2} - 755673498480 T^{3} + 11432646907974436 T^{4} - 93264987784186265040 T^{5} + \)\(54\!\cdots\!00\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(41\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!60\)\( p^{4} T^{9} + \)\(54\!\cdots\!00\)\( p^{8} T^{10} - 93264987784186265040 p^{12} T^{11} + 11432646907974436 p^{16} T^{12} - 755673498480 p^{20} T^{13} + 80899200 p^{24} T^{14} - 12720 p^{28} T^{15} + p^{32} T^{16} \) |
| 89 | \( 1 - 389499328 T^{2} + 72337503784652668 T^{4} - \)\(82\!\cdots\!76\)\( T^{6} + \)\(62\!\cdots\!70\)\( T^{8} - \)\(82\!\cdots\!76\)\( p^{8} T^{10} + 72337503784652668 p^{16} T^{12} - 389499328 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( 1 + 15600 T + 121680000 T^{2} + 1725033190800 T^{3} + 23516472780776956 T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(85\!\cdots\!00\)\( T^{6} + \)\(88\!\cdots\!00\)\( T^{7} + \)\(89\!\cdots\!26\)\( T^{8} + \)\(88\!\cdots\!00\)\( p^{4} T^{9} + \)\(85\!\cdots\!00\)\( p^{8} T^{10} + \)\(14\!\cdots\!00\)\( p^{12} T^{11} + 23516472780776956 p^{16} T^{12} + 1725033190800 p^{20} T^{13} + 121680000 p^{24} T^{14} + 15600 p^{28} T^{15} + p^{32} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50238938461038962088018973641, −6.23512163985146464946271264412, −6.12143641299455294372560664182, −6.04310913942516550112456851321, −5.83931068604989715082959404815, −5.33492189498502780143639211037, −5.21239583927455857471810146384, −5.06793063842064826217149550712, −4.72528322417607629606663158475, −4.59929043636936695657432808011, −4.20114517711439087711634581677, −3.92026165563827639042486941628, −3.87940239872145145947649045004, −3.79584262259532360228279914938, −3.57030510345658695220234257061, −3.07404111189400603123428895567, −2.91849018054907592396123581487, −2.77130912119517751410026457508, −2.56978428598701003118810173891, −1.82165442790926178713052259981, −1.76821654879350274912546019056, −1.47088944716939243370426519827, −1.04498397352996026039602566346, −0.53093047810948250745902207118, −0.17739548307531552967008497018,
0.17739548307531552967008497018, 0.53093047810948250745902207118, 1.04498397352996026039602566346, 1.47088944716939243370426519827, 1.76821654879350274912546019056, 1.82165442790926178713052259981, 2.56978428598701003118810173891, 2.77130912119517751410026457508, 2.91849018054907592396123581487, 3.07404111189400603123428895567, 3.57030510345658695220234257061, 3.79584262259532360228279914938, 3.87940239872145145947649045004, 3.92026165563827639042486941628, 4.20114517711439087711634581677, 4.59929043636936695657432808011, 4.72528322417607629606663158475, 5.06793063842064826217149550712, 5.21239583927455857471810146384, 5.33492189498502780143639211037, 5.83931068604989715082959404815, 6.04310913942516550112456851321, 6.12143641299455294372560664182, 6.23512163985146464946271264412, 6.50238938461038962088018973641
Plot not available for L-functions of degree greater than 10.