| L(s) = 1 | + 52·7-s + 220·13-s − 308·19-s + 2.33e3·25-s − 1.47e3·31-s + 1.22e3·37-s + 388·43-s − 6.53e3·49-s + 1.80e3·61-s − 2.18e3·67-s − 4.54e3·73-s − 5.01e3·79-s + 1.14e4·91-s − 2.52e3·97-s + 1.61e4·103-s + 1.10e4·109-s + 5.10e4·121-s + 127-s + 131-s − 1.60e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.06·7-s + 1.30·13-s − 0.853·19-s + 3.73·25-s − 1.53·31-s + 0.897·37-s + 0.209·43-s − 2.72·49-s + 0.484·61-s − 0.485·67-s − 0.852·73-s − 0.803·79-s + 1.38·91-s − 0.268·97-s + 1.52·103-s + 0.931·109-s + 3.48·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.905·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.848476164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.848476164\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2336 T^{2} + 136558 p^{2} T^{4} - 658628608 p T^{6} + 2404523790019 T^{8} - 658628608 p^{9} T^{10} + 136558 p^{18} T^{12} - 2336 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 - 26 T + 4282 T^{2} - 214310 T^{3} + 11854978 T^{4} - 214310 p^{4} T^{5} + 4282 p^{8} T^{6} - 26 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 - 51068 T^{2} + 1609814488 T^{4} - 35017149442772 T^{6} + 585430682946749806 T^{8} - 35017149442772 p^{8} T^{10} + 1609814488 p^{16} T^{12} - 51068 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 110 T + 50350 T^{2} - 3749312 T^{3} + 1512180355 T^{4} - 3749312 p^{4} T^{5} + 50350 p^{8} T^{6} - 110 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 581912 T^{2} + 154148002822 T^{4} - 84008052201728 p^{2} T^{6} + 29760879019902523 p^{4} T^{8} - 84008052201728 p^{10} T^{10} + 154148002822 p^{16} T^{12} - 581912 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 + 154 T + 428050 T^{2} + 50927398 T^{3} + 78431447410 T^{4} + 50927398 p^{4} T^{5} + 428050 p^{8} T^{6} + 154 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1539572 T^{2} + 1071515224168 T^{4} - 464417540343703388 T^{6} + \)\(14\!\cdots\!66\)\( T^{8} - 464417540343703388 p^{8} T^{10} + 1071515224168 p^{16} T^{12} - 1539572 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 1716392 T^{2} + 2128658111734 T^{4} - 1990766770969759616 T^{6} + \)\(16\!\cdots\!35\)\( T^{8} - 1990766770969759616 p^{8} T^{10} + 2128658111734 p^{16} T^{12} - 1716392 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 736 T + 1540120 T^{2} + 1931471584 T^{3} + 1422185749294 T^{4} + 1931471584 p^{4} T^{5} + 1540120 p^{8} T^{6} + 736 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 614 T + 4316302 T^{2} - 1557229880 T^{3} + 8895611838523 T^{4} - 1557229880 p^{4} T^{5} + 4316302 p^{8} T^{6} - 614 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 3947288 T^{2} - 3616850163692 T^{4} + 16457572376333935288 T^{6} + \)\(20\!\cdots\!26\)\( T^{8} + 16457572376333935288 p^{8} T^{10} - 3616850163692 p^{16} T^{12} - 3947288 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 - 194 T + 5548210 T^{2} + 7089791362 T^{3} + 12339731626930 T^{4} + 7089791362 p^{4} T^{5} + 5548210 p^{8} T^{6} - 194 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 4274840 T^{2} + 1700192516276 p T^{4} - \)\(29\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!36\)\( p^{8} T^{10} + 1700192516276 p^{17} T^{12} - 4274840 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 6401528 T^{2} + 78679698713236 T^{4} - \)\(35\!\cdots\!72\)\( T^{6} + \)\(43\!\cdots\!02\)\( T^{8} - \)\(35\!\cdots\!72\)\( p^{8} T^{10} + 78679698713236 p^{16} T^{12} - 6401528 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 12582392 T^{2} + 342351109975372 T^{4} - \)\(46\!\cdots\!08\)\( T^{6} + \)\(57\!\cdots\!38\)\( T^{8} - \)\(46\!\cdots\!08\)\( p^{8} T^{10} + 342351109975372 p^{16} T^{12} - 12582392 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 902 T + 6189730 T^{2} + 477659584 p T^{3} - 232110610634705 T^{4} + 477659584 p^{5} T^{5} + 6189730 p^{8} T^{6} - 902 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 1090 T + 32366794 T^{2} + 11878062622 T^{3} + 477344036370850 T^{4} + 11878062622 p^{4} T^{5} + 32366794 p^{8} T^{6} + 1090 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 15796628 T^{2} + 1499316890690728 T^{4} - \)\(15\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!18\)\( T^{8} - \)\(15\!\cdots\!88\)\( p^{8} T^{10} + 1499316890690728 p^{16} T^{12} - 15796628 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 + 2272 T + 39410902 T^{2} - 196772776832 T^{3} + 103134174803371 T^{4} - 196772776832 p^{4} T^{5} + 39410902 p^{8} T^{6} + 2272 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2506 T + 105092818 T^{2} + 227234889574 T^{3} + 5831277891584050 T^{4} + 227234889574 p^{4} T^{5} + 105092818 p^{8} T^{6} + 2506 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 175659320 T^{2} + 14988116577681148 T^{4} - \)\(80\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!38\)\( T^{8} - \)\(80\!\cdots\!00\)\( p^{8} T^{10} + 14988116577681148 p^{16} T^{12} - 175659320 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 259833392 T^{2} + 39708026833239838 T^{4} - \)\(39\!\cdots\!88\)\( T^{6} + \)\(29\!\cdots\!31\)\( T^{8} - \)\(39\!\cdots\!88\)\( p^{8} T^{10} + 39708026833239838 p^{16} T^{12} - 259833392 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 1264 T + 258571096 T^{2} + 226571051536 T^{3} + 31287418967396782 T^{4} + 226571051536 p^{4} T^{5} + 258571096 p^{8} T^{6} + 1264 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−4.37059870647163855486639299353, −4.33544912025161010441534066542, −4.20316461100393495306890545548, −4.10920032790796118368755690814, −3.83914539512597557198157772260, −3.71913971554388558444404196196, −3.46038573563215875294477816037, −3.28970226991722105504724541109, −3.17139514966467069883175681050, −3.08935197795839750097216478724, −3.01277375738216796104785418189, −2.74331058426218025477431152449, −2.43671908103018673795769465006, −2.42184164667001601525785048229, −2.04062514935206605908658710614, −1.88407353850612033289210908917, −1.86596693403454774806192085057, −1.56152071142343207297921230078, −1.43008940036204012333934124375, −1.16432430766778003789324496080, −0.979413586458865241180524769233, −0.820376363515355106686864354546, −0.73747563038310040198695309883, −0.26384019053372859666262456582, −0.18536102201605698400655128128,
0.18536102201605698400655128128, 0.26384019053372859666262456582, 0.73747563038310040198695309883, 0.820376363515355106686864354546, 0.979413586458865241180524769233, 1.16432430766778003789324496080, 1.43008940036204012333934124375, 1.56152071142343207297921230078, 1.86596693403454774806192085057, 1.88407353850612033289210908917, 2.04062514935206605908658710614, 2.42184164667001601525785048229, 2.43671908103018673795769465006, 2.74331058426218025477431152449, 3.01277375738216796104785418189, 3.08935197795839750097216478724, 3.17139514966467069883175681050, 3.28970226991722105504724541109, 3.46038573563215875294477816037, 3.71913971554388558444404196196, 3.83914539512597557198157772260, 4.10920032790796118368755690814, 4.20316461100393495306890545548, 4.33544912025161010441534066542, 4.37059870647163855486639299353
Plot not available for L-functions of degree greater than 10.
