| L(s) = 1 | − 64·4-s − 729·9-s + 1.46e4·11-s + 4.09e3·16-s − 4.97e4·19-s + 8.32e4·29-s + 6.63e4·31-s + 4.66e4·36-s − 1.27e6·41-s − 9.38e5·44-s − 8.36e5·49-s − 1.49e6·59-s − 3.32e6·61-s − 2.62e5·64-s + 1.14e7·71-s + 3.18e6·76-s − 7.61e6·79-s + 5.31e5·81-s − 1.19e7·89-s − 1.06e7·99-s − 3.45e7·101-s + 1.76e6·109-s − 5.32e6·116-s + 1.22e8·121-s − 4.24e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 3.32·11-s + 1/4·16-s − 1.66·19-s + 0.633·29-s + 0.399·31-s + 1/6·36-s − 2.89·41-s − 1.66·44-s − 1.01·49-s − 0.944·59-s − 1.87·61-s − 1/8·64-s + 3.79·71-s + 0.831·76-s − 1.73·79-s + 1/9·81-s − 1.80·89-s − 1.10·99-s − 3.33·101-s + 0.130·109-s − 0.316·116-s + 6.27·121-s − 0.199·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.800349123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800349123\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 836690 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 7332 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 111041830 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 777038110 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 24860 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5091714190 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 41610 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 33152 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 188533985110 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 639078 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 519172508470 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 825084622750 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1731503657590 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 745140 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1660618 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1291821631750 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5716152 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 15019739667790 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3807440 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 49301574416230 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5991210 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145111945820350 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.94535706495741052907247047185, −11.40984217045861087073177300620, −11.17391826632073922182688519406, −10.27191364918195846728810063452, −9.892707033223472850539622348479, −9.110407283549071592264365201340, −9.052436518910915493277285404491, −8.399296501337483566984073223979, −8.001926788436808073557100539542, −6.81947149918552834108133314616, −6.52972263690051921750906443345, −6.37821429616824592923845155957, −5.35945466630137430125207921113, −4.61493906189933845358163443057, −4.04558705313460463098575276530, −3.69613564089912138883815512913, −2.84433024435089193786836395804, −1.55689069531432769278551947151, −1.49122033639136064687903721079, −0.37819288459320960223283604476,
0.37819288459320960223283604476, 1.49122033639136064687903721079, 1.55689069531432769278551947151, 2.84433024435089193786836395804, 3.69613564089912138883815512913, 4.04558705313460463098575276530, 4.61493906189933845358163443057, 5.35945466630137430125207921113, 6.37821429616824592923845155957, 6.52972263690051921750906443345, 6.81947149918552834108133314616, 8.001926788436808073557100539542, 8.399296501337483566984073223979, 9.052436518910915493277285404491, 9.110407283549071592264365201340, 9.892707033223472850539622348479, 10.27191364918195846728810063452, 11.17391826632073922182688519406, 11.40984217045861087073177300620, 11.94535706495741052907247047185
