L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 7-s − 2·9-s − 2·10-s + 11-s + 2·12-s + 13-s − 2·14-s − 15-s − 4·16-s + 3·17-s − 4·18-s − 4·19-s − 2·20-s − 21-s + 2·22-s + 4·23-s + 25-s + 2·26-s − 5·27-s − 2·28-s + 3·29-s − 2·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s − 16-s + 0.727·17-s − 0.942·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.377·28-s + 0.557·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.70579896069875, −14.31842844136945, −13.84837980251427, −13.30612635854217, −12.95017738883197, −12.28864907034558, −11.97142465963582, −11.41292451850247, −10.89816010522127, −10.27651850654239, −9.549919779126432, −8.914405272135311, −8.593379451319113, −7.963072689267273, −7.264174383835884, −6.656713562043143, −6.140315491400020, −5.639126128333161, −4.976242807556482, −4.360699244432827, −3.817560571891851, −3.261557652024564, −2.810758018905363, −2.207686478795953, −1.087696937401822, 0,
1.087696937401822, 2.207686478795953, 2.810758018905363, 3.261557652024564, 3.817560571891851, 4.360699244432827, 4.976242807556482, 5.639126128333161, 6.140315491400020, 6.656713562043143, 7.264174383835884, 7.963072689267273, 8.593379451319113, 8.914405272135311, 9.549919779126432, 10.27651850654239, 10.89816010522127, 11.41292451850247, 11.97142465963582, 12.28864907034558, 12.95017738883197, 13.30612635854217, 13.84837980251427, 14.31842844136945, 14.70579896069875