| L(s) = 1 | − 8.05·2-s − 447.·4-s − 625·5-s − 2.40e3·7-s + 7.72e3·8-s + 5.03e3·10-s + 4.77e4·11-s − 7.12e4·13-s + 1.93e4·14-s + 1.66e5·16-s + 4.33e4·17-s − 4.63e5·19-s + 2.79e5·20-s − 3.84e5·22-s + 5.02e5·23-s + 3.90e5·25-s + 5.73e5·26-s + 1.07e6·28-s + 3.95e6·29-s + 1.81e5·31-s − 5.29e6·32-s − 3.49e5·34-s + 1.50e6·35-s − 1.74e7·37-s + 3.73e6·38-s − 4.82e6·40-s − 3.40e7·41-s + ⋯ |
| L(s) = 1 | − 0.355·2-s − 0.873·4-s − 0.447·5-s − 0.377·7-s + 0.666·8-s + 0.159·10-s + 0.982·11-s − 0.691·13-s + 0.134·14-s + 0.636·16-s + 0.125·17-s − 0.816·19-s + 0.390·20-s − 0.349·22-s + 0.374·23-s + 0.200·25-s + 0.246·26-s + 0.330·28-s + 1.03·29-s + 0.0353·31-s − 0.893·32-s − 0.0448·34-s + 0.169·35-s − 1.53·37-s + 0.290·38-s − 0.298·40-s − 1.88·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7767643684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7767643684\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 8.05T + 512T^{2} \) |
| 11 | \( 1 - 4.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.12e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.33e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.63e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.02e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.95e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.81e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.74e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.40e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.41e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.60e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.04e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.98e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.57e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.84e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.06e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.06e8T + 7.60e17T^{2} \) |
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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
−9.971827047736543396213576789061, −9.099555159852882537313652709777, −8.424584241122893216747432376033, −7.35094885444320571734963509662, −6.37983521700026073231362225020, −5.00688258548392627113249146455, −4.18140030070642521954758674095, −3.16801150812862523205257615068, −1.57709836941387653183156151435, −0.42434586945998333900352262217,
0.42434586945998333900352262217, 1.57709836941387653183156151435, 3.16801150812862523205257615068, 4.18140030070642521954758674095, 5.00688258548392627113249146455, 6.37983521700026073231362225020, 7.35094885444320571734963509662, 8.424584241122893216747432376033, 9.099555159852882537313652709777, 9.971827047736543396213576789061
