| L(s) = 1 | + 16·2-s + 256·4-s − 625·5-s + 5.43e3·7-s + 4.09e3·8-s − 1.00e4·10-s − 7.39e4·11-s − 1.14e5·13-s + 8.69e4·14-s + 6.55e4·16-s − 4.16e4·17-s + 1.05e6·19-s − 1.60e5·20-s − 1.18e6·22-s − 1.59e6·23-s + 3.90e5·25-s − 1.83e6·26-s + 1.39e6·28-s − 2.18e6·29-s − 9.61e6·31-s + 1.04e6·32-s − 6.66e5·34-s − 3.39e6·35-s + 4.79e6·37-s + 1.69e7·38-s − 2.56e6·40-s − 9.53e6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.855·7-s + 0.353·8-s − 0.316·10-s − 1.52·11-s − 1.11·13-s + 0.604·14-s + 1/4·16-s − 0.121·17-s + 1.86·19-s − 0.223·20-s − 1.07·22-s − 1.19·23-s + 1/5·25-s − 0.786·26-s + 0.427·28-s − 0.573·29-s − 1.87·31-s + 0.176·32-s − 0.0855·34-s − 0.382·35-s + 0.421·37-s + 1.31·38-s − 0.158·40-s − 0.526·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{4} T \) |
| good | 7 | \( 1 - 776 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 73932 T + p^{9} T^{2} \) |
| 13 | \( 1 + 114514 T + p^{9} T^{2} \) |
| 17 | \( 1 + 41682 T + p^{9} T^{2} \) |
| 19 | \( 1 - 1057460 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1599336 T + p^{9} T^{2} \) |
| 29 | \( 1 + 2184510 T + p^{9} T^{2} \) |
| 31 | \( 1 + 9619648 T + p^{9} T^{2} \) |
| 37 | \( 1 - 4799942 T + p^{9} T^{2} \) |
| 41 | \( 1 + 9531882 T + p^{9} T^{2} \) |
| 43 | \( 1 + 13464484 T + p^{9} T^{2} \) |
| 47 | \( 1 + 11441952 T + p^{9} T^{2} \) |
| 53 | \( 1 + 53615766 T + p^{9} T^{2} \) |
| 59 | \( 1 + 81862620 T + p^{9} T^{2} \) |
| 61 | \( 1 + 104691298 T + p^{9} T^{2} \) |
| 67 | \( 1 - 2098076 p T + p^{9} T^{2} \) |
| 71 | \( 1 + 97098792 T + p^{9} T^{2} \) |
| 73 | \( 1 - 171848906 T + p^{9} T^{2} \) |
| 79 | \( 1 + 117380080 T + p^{9} T^{2} \) |
| 83 | \( 1 + 323637636 T + p^{9} T^{2} \) |
| 89 | \( 1 - 894379110 T + p^{9} T^{2} \) |
| 97 | \( 1 - 232678562 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82336980309008928478187263927, −10.93114075896393162623737905050, −9.720896627614157900209634536857, −7.929698469879718992089837176458, −7.39612062097473017917133447459, −5.50923806667258108381752824445, −4.77217715538688757939536866526, −3.23679421061112865155388355762, −1.90903220702773943085929724147, 0,
1.90903220702773943085929724147, 3.23679421061112865155388355762, 4.77217715538688757939536866526, 5.50923806667258108381752824445, 7.39612062097473017917133447459, 7.929698469879718992089837176458, 9.720896627614157900209634536857, 10.93114075896393162623737905050, 11.82336980309008928478187263927
