| L(s) = 1 | + 24·3-s + 4·7-s + 324·9-s + 56·19-s + 96·21-s + 172·25-s + 3.24e3·27-s + 240·29-s − 320·31-s + 392·37-s + 816·47-s + 288·53-s + 1.34e3·57-s + 1.82e3·59-s + 1.29e3·63-s + 4.12e3·75-s + 2.67e4·81-s − 1.68e3·83-s + 5.76e3·87-s − 7.68e3·93-s − 3.58e3·103-s − 2.62e3·109-s + 9.40e3·111-s + 4.70e3·113-s + 7.68e3·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 0.215·7-s + 12·9-s + 0.676·19-s + 0.997·21-s + 1.37·25-s + 23.0·27-s + 1.53·29-s − 1.85·31-s + 1.74·37-s + 2.53·47-s + 0.746·53-s + 3.12·57-s + 4.02·59-s + 2.59·63-s + 6.35·75-s + 36.6·81-s − 2.22·83-s + 7.09·87-s − 8.56·93-s − 3.42·103-s − 2.30·109-s + 8.04·111-s + 3.91·113-s + 5.77·121-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(274.4436596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(274.4436596\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T )^{8} \) |
| 7 | \( 1 - 4 T + 16 T^{2} + 436 p T^{3} - 3490 p^{2} T^{4} + 436 p^{4} T^{5} + 16 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| good | 5 | \( 1 - 172 T^{2} + 12088 T^{4} - 1922884 T^{6} + 432877966 T^{8} - 1922884 p^{6} T^{10} + 12088 p^{12} T^{12} - 172 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 7684 T^{2} + 29103112 T^{4} - 68711357356 T^{6} + 109991929594990 T^{8} - 68711357356 p^{6} T^{10} + 29103112 p^{12} T^{12} - 7684 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( 1 - 7736 T^{2} + 33258844 T^{4} - 103335049928 T^{6} + 251087208174118 T^{8} - 103335049928 p^{6} T^{10} + 33258844 p^{12} T^{12} - 7736 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 25708 T^{2} + 341684344 T^{4} - 2888689279012 T^{6} + 16925607763770286 T^{8} - 2888689279012 p^{6} T^{10} + 341684344 p^{12} T^{12} - 25708 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 28 T + 17056 T^{2} - 75836 T^{3} + 133946894 T^{4} - 75836 p^{3} T^{5} + 17056 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 29476 T^{2} + 511325128 T^{4} - 7961603563564 T^{6} + 110519492921771470 T^{8} - 7961603563564 p^{6} T^{10} + 511325128 p^{12} T^{12} - 29476 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 - 120 T + 65660 T^{2} - 6712488 T^{3} + 2255693046 T^{4} - 6712488 p^{3} T^{5} + 65660 p^{6} T^{6} - 120 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 160 T + 36172 T^{2} - 6186976 T^{3} - 728821786 T^{4} - 6186976 p^{3} T^{5} + 36172 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 196 T + 70696 T^{2} - 15303292 T^{3} + 2331128254 T^{4} - 15303292 p^{3} T^{5} + 70696 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( 1 - 237772 T^{2} + 34664599288 T^{4} - 3564171608723332 T^{6} + \)\(28\!\cdots\!46\)\( T^{8} - 3564171608723332 p^{6} T^{10} + 34664599288 p^{12} T^{12} - 237772 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( 1 - 273872 T^{2} + 34345238716 T^{4} - 2691741399065264 T^{6} + \)\(18\!\cdots\!22\)\( T^{8} - 2691741399065264 p^{6} T^{10} + 34345238716 p^{12} T^{12} - 273872 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 - 408 T + 266828 T^{2} - 62051256 T^{3} + 32091577638 T^{4} - 62051256 p^{3} T^{5} + 266828 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 144 T + 272156 T^{2} - 93836016 T^{3} + 36246347862 T^{4} - 93836016 p^{3} T^{5} + 272156 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 912 T + 752684 T^{2} - 485643024 T^{3} + 219867046134 T^{4} - 485643024 p^{3} T^{5} + 752684 p^{6} T^{6} - 912 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( 1 - 1695272 T^{2} + 1282321093564 T^{4} - 565249437068217176 T^{6} + \)\(15\!\cdots\!66\)\( T^{8} - 565249437068217176 p^{6} T^{10} + 1282321093564 p^{12} T^{12} - 1695272 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( 1 - 1674176 T^{2} + 1368230434588 T^{4} - 709074585714612800 T^{6} + \)\(25\!\cdots\!78\)\( T^{8} - 709074585714612800 p^{6} T^{10} + 1368230434588 p^{12} T^{12} - 1674176 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( 1 - 1345636 T^{2} + 1085567531464 T^{4} - 599054164045845100 T^{6} + \)\(24\!\cdots\!66\)\( T^{8} - 599054164045845100 p^{6} T^{10} + 1085567531464 p^{12} T^{12} - 1345636 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( 1 - 1739480 T^{2} + 1687675343356 T^{4} - 1071753557470214888 T^{6} + \)\(49\!\cdots\!98\)\( T^{8} - 1071753557470214888 p^{6} T^{10} + 1687675343356 p^{12} T^{12} - 1739480 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( 1 - 3231392 T^{2} + 4863579897916 T^{4} - 4425141721050671456 T^{6} + \)\(26\!\cdots\!14\)\( T^{8} - 4425141721050671456 p^{6} T^{10} + 4863579897916 p^{12} T^{12} - 3231392 p^{18} T^{14} + p^{24} T^{16} \) |
| 83 | \( ( 1 + 840 T + 2002748 T^{2} + 1107399240 T^{3} + 1578867394614 T^{4} + 1107399240 p^{3} T^{5} + 2002748 p^{6} T^{6} + 840 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 - 3966028 T^{2} + 7744575214072 T^{4} - 9543109070842050436 T^{6} + \)\(80\!\cdots\!46\)\( T^{8} - 9543109070842050436 p^{6} T^{10} + 7744575214072 p^{12} T^{12} - 3966028 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( 1 - 3256088 T^{2} + 6152309237692 T^{4} - 8283569126499185960 T^{6} + \)\(84\!\cdots\!90\)\( T^{8} - 8283569126499185960 p^{6} T^{10} + 6152309237692 p^{12} T^{12} - 3256088 p^{18} T^{14} + p^{24} T^{16} \) |
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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
−4.47871119136962958258625529073, −4.22746930016899700361751385580, −4.19987069216158397640417053963, −4.04343946812374810157791322688, −4.03520344343480483389420040213, −3.82954382350319618834055332991, −3.73700585890821500088757460687, −3.63768503894790659796903283062, −3.36269727769005325578079853500, −2.95871216016229305772206038533, −2.93801542021059647516561925419, −2.88972673880530299538447841126, −2.74952431730462316477991205190, −2.67164288986774622267990867864, −2.63903669584819798090811367675, −2.23603043986500777417379887194, −1.92102480922885405489974279177, −1.91066099353307735198466341630, −1.81608843875096573304483823803, −1.45685784158905960381508891260, −1.42729291816592978989695321653, −0.840922525720439216495731065406, −0.810201416041052192279039927059, −0.64196891002919124723739912887, −0.53269738532334219620332243955,
0.53269738532334219620332243955, 0.64196891002919124723739912887, 0.810201416041052192279039927059, 0.840922525720439216495731065406, 1.42729291816592978989695321653, 1.45685784158905960381508891260, 1.81608843875096573304483823803, 1.91066099353307735198466341630, 1.92102480922885405489974279177, 2.23603043986500777417379887194, 2.63903669584819798090811367675, 2.67164288986774622267990867864, 2.74952431730462316477991205190, 2.88972673880530299538447841126, 2.93801542021059647516561925419, 2.95871216016229305772206038533, 3.36269727769005325578079853500, 3.63768503894790659796903283062, 3.73700585890821500088757460687, 3.82954382350319618834055332991, 4.03520344343480483389420040213, 4.04343946812374810157791322688, 4.19987069216158397640417053963, 4.22746930016899700361751385580, 4.47871119136962958258625529073
Plot not available for L-functions of degree greater than 10.
