| L(s) = 1 | + 32·2-s + 114·3-s + 512·4-s + 5.85e3·5-s + 3.64e3·6-s + 1.39e4·7-s + 6.49e3·9-s + 1.87e5·10-s + 1.50e5·11-s + 5.83e4·12-s + 2.19e5·13-s + 4.44e5·14-s + 6.66e5·15-s − 2.62e5·16-s − 3.05e6·17-s + 2.07e5·18-s + 2.99e6·20-s + 1.58e6·21-s + 4.81e6·22-s − 1.42e6·23-s + 2.44e7·25-s + 7.03e6·26-s + 6.73e6·27-s + 7.11e6·28-s + 2.13e7·30-s − 5.81e7·31-s − 8.38e6·32-s + ⋯ |
| L(s) = 1 | + 2-s + 0.469·3-s + 1/2·4-s + 1.87·5-s + 0.469·6-s + 0.827·7-s + 0.110·9-s + 1.87·10-s + 0.934·11-s + 0.234·12-s + 0.591·13-s + 0.827·14-s + 0.878·15-s − 1/4·16-s − 2.15·17-s + 0.110·18-s + 0.935·20-s + 0.388·21-s + 0.934·22-s − 0.221·23-s + 2.50·25-s + 0.591·26-s + 0.469·27-s + 0.413·28-s + 0.878·30-s − 2.03·31-s − 1/4·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(6.355459029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.355459029\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 - p^{5} T + p^{9} T^{2} \) |
| 5 | $C_2$ | \( 1 - 234 p^{2} T + p^{10} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 p T + 722 p^{2} T^{2} - 38 p^{11} T^{3} + p^{20} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13906 T + 96688418 T^{2} - 13906 p^{10} T^{3} + p^{20} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 75242 T + p^{10} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 219714 T + 24137120898 T^{2} - 219714 p^{10} T^{3} + p^{20} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3057854 T + 4675235542658 T^{2} + 3057854 p^{10} T^{3} + p^{20} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4048803626798 T^{2} + p^{20} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1424846 T + 1015093061858 T^{2} + 1424846 p^{10} T^{3} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 841215443546002 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 29080718 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 1823694 T + 1662929902818 T^{2} + 1823694 p^{10} T^{3} + p^{20} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 163945678 T + p^{10} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 236845554 T + 28047908224783458 T^{2} - 236845554 p^{10} T^{3} + p^{20} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 552640626 T + 152705830752835938 T^{2} - 552640626 p^{10} T^{3} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 616920194 T + 190295262882498818 T^{2} - 616920194 p^{10} T^{3} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 136962600617793202 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1353610038 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1707141826 T + 1457166607039307138 T^{2} - 1707141826 p^{10} T^{3} + p^{20} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2827014562 T + p^{10} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5506594366 T + 15161290755831470978 T^{2} + 5506594366 p^{10} T^{3} + p^{20} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7884247659893284802 T^{2} + p^{20} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2692678194 T + 3625257928221550818 T^{2} - 2692678194 p^{10} T^{3} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 55248148650264264802 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1053125694 T + 554536863681490818 T^{2} + 1053125694 p^{10} T^{3} + p^{20} T^{4} \) |
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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
−18.39634524542139402580114584633, −18.24201118133087810164147784822, −17.26944608037313628008584437461, −16.89361649096381474158077676476, −15.60467157306474261588579157493, −14.91836961386299830575491805189, −14.11076234856294098963190874907, −13.78833395774184528742258799633, −13.21320892687923712486291325465, −12.32121513427175594119994418304, −11.14695353300958144043870223346, −10.48187388561386472360744690374, −8.987284424356253584939278611471, −8.948689522200678052404287025130, −6.98215040703774643879951698499, −6.13260724991758774708242829790, −5.16464369333576426631803464446, −4.02374903709250392214806328264, −2.37842261372478713179066510225, −1.58099731909072381970786310110,
1.58099731909072381970786310110, 2.37842261372478713179066510225, 4.02374903709250392214806328264, 5.16464369333576426631803464446, 6.13260724991758774708242829790, 6.98215040703774643879951698499, 8.948689522200678052404287025130, 8.987284424356253584939278611471, 10.48187388561386472360744690374, 11.14695353300958144043870223346, 12.32121513427175594119994418304, 13.21320892687923712486291325465, 13.78833395774184528742258799633, 14.11076234856294098963190874907, 14.91836961386299830575491805189, 15.60467157306474261588579157493, 16.89361649096381474158077676476, 17.26944608037313628008584437461, 18.24201118133087810164147784822, 18.39634524542139402580114584633
