| L(s) = 1 | − 497·4-s − 1.31e4·9-s + 7.09e4·11-s + 8.77e4·16-s + 8.06e5·19-s + 1.49e5·29-s − 1.00e7·31-s + 6.52e6·36-s + 2.84e7·41-s − 3.52e7·44-s + 8.35e7·49-s − 3.75e8·59-s + 3.08e8·61-s − 9.47e7·64-s − 4.56e8·71-s − 4.00e8·76-s + 1.86e9·79-s + 1.29e8·81-s − 4.49e8·89-s − 9.31e8·99-s + 6.98e8·101-s + 5.89e9·109-s − 7.41e7·116-s + 2.72e9·121-s + 4.99e9·124-s + ⋯ |
| L(s) = 1 | − 0.970·4-s − 2/3·9-s + 1.46·11-s + 0.334·16-s + 1.41·19-s + 0.0391·29-s − 1.95·31-s + 0.647·36-s + 1.57·41-s − 1.41·44-s + 2.06·49-s − 4.03·59-s + 2.84·61-s − 0.705·64-s − 2.13·71-s − 1.37·76-s + 5.38·79-s + 1/3·81-s − 0.758·89-s − 0.974·99-s + 0.667·101-s + 3.99·109-s − 0.0380·116-s + 1.15·121-s + 1.89·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.476327501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.476327501\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 497 T^{2} + 9953 p^{4} T^{4} + 497 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1704636 p^{2} T^{2} + 38066346502 p^{6} T^{4} - 1704636 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 35488 T + 526013014 T^{2} - 35488 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 27567505292 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - 27567505292 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 388734753820 T^{2} + \)\(65\!\cdots\!18\)\( T^{4} - 388734753820 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 403296 T + 684929514838 T^{2} - 403296 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2800589695204 T^{2} + \)\(81\!\cdots\!58\)\( T^{4} + 2800589695204 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 74572 T + 28833430018078 T^{2} - 74572 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5027128 T + 52415931233342 T^{2} + 5027128 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 433247572021484 T^{2} + \)\(79\!\cdots\!78\)\( T^{4} - 433247572021484 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 14211332 T + 443988635955862 T^{2} - 14211332 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1570463388232460 T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - 1570463388232460 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 568239172667780 T^{2} + \)\(55\!\cdots\!78\)\( T^{4} + 568239172667780 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9086956380170956 T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - 9086956380170956 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 187863136 T + 23071633420288438 T^{2} + 187863136 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 154080060 T + 23302683905802238 T^{2} - 154080060 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22768078838174100 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + 22768078838174100 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 228270976 T + 45777616900481806 T^{2} + 228270976 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 135645128086811996 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - 135645128086811996 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 932406760 T + 453226630902929438 T^{2} - 932406760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 702700996120787372 T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - 702700996120787372 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 2522676 p T + 610925899926766678 T^{2} + 2522676 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1619811253917303940 T^{2} + \)\(14\!\cdots\!78\)\( T^{4} + 1619811253917303940 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.861995564043158600550407689141, −8.797115523599284500629425949142, −8.328464809098468629459041243860, −7.82595793625167115516015003601, −7.54361226557208892906640073637, −7.31516435750804656014312994521, −7.26892500999050496264216908578, −6.29693539090020827415507835502, −6.27049299543984017842672508486, −6.23674699588393933795335065298, −5.48817746286961414954993311600, −5.18696901058967816579471242170, −5.17485839969695067942464196411, −4.53243215947305278474614437816, −4.17717680240127845938505553755, −3.77545806245053456375930399975, −3.73021942516890110223204600382, −3.00052107820511371635171061409, −2.98786495756296376695435019039, −2.24012796943732581602349303239, −1.81531133731932297085009611361, −1.50626965569955646578854619747, −0.876751785216445830206569210518, −0.62942316954173955887747653092, −0.39868963713341195395434252827,
0.39868963713341195395434252827, 0.62942316954173955887747653092, 0.876751785216445830206569210518, 1.50626965569955646578854619747, 1.81531133731932297085009611361, 2.24012796943732581602349303239, 2.98786495756296376695435019039, 3.00052107820511371635171061409, 3.73021942516890110223204600382, 3.77545806245053456375930399975, 4.17717680240127845938505553755, 4.53243215947305278474614437816, 5.17485839969695067942464196411, 5.18696901058967816579471242170, 5.48817746286961414954993311600, 6.23674699588393933795335065298, 6.27049299543984017842672508486, 6.29693539090020827415507835502, 7.26892500999050496264216908578, 7.31516435750804656014312994521, 7.54361226557208892906640073637, 7.82595793625167115516015003601, 8.328464809098468629459041243860, 8.797115523599284500629425949142, 8.861995564043158600550407689141
