| L(s) = 1 | + 38.8·2-s + 996.·4-s + 625·5-s + 2.40e3·7-s + 1.88e4·8-s + 2.42e4·10-s − 6.97e3·11-s + 4.56e4·13-s + 9.32e4·14-s + 2.20e5·16-s + 1.54e5·17-s − 3.80e5·19-s + 6.22e5·20-s − 2.70e5·22-s + 1.66e6·23-s + 3.90e5·25-s + 1.77e6·26-s + 2.39e6·28-s + 2.23e6·29-s + 5.92e6·31-s − 1.06e6·32-s + 6.01e6·34-s + 1.50e6·35-s + 4.08e6·37-s − 1.47e7·38-s + 1.17e7·40-s − 1.62e6·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.94·4-s + 0.447·5-s + 0.377·7-s + 1.62·8-s + 0.767·10-s − 0.143·11-s + 0.442·13-s + 0.648·14-s + 0.841·16-s + 0.449·17-s − 0.670·19-s + 0.870·20-s − 0.246·22-s + 1.24·23-s + 0.200·25-s + 0.760·26-s + 0.735·28-s + 0.586·29-s + 1.15·31-s − 0.180·32-s + 0.771·34-s + 0.169·35-s + 0.358·37-s − 1.15·38-s + 0.726·40-s − 0.0895·41-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
| \(L(5)\) |
\(\approx\) |
\(8.807067523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.807067523\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 38.8T + 512T^{2} \) |
| 11 | \( 1 + 6.97e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.56e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.54e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.80e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.66e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.08e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.62e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.61e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.12e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.31e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.91e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.51e8T + 7.60e17T^{2} \) |
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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
−10.54467868267063436405676907601, −9.195120008723961496640752713863, −7.965626980883915701642420089055, −6.77343651103215939858800134196, −6.01668921505784330015371331683, −5.08668982455218909441288772527, −4.31136069549469982418368994431, −3.16210359405961876123728902440, −2.28577097016992627238319050224, −1.05249568027173207308618487084,
1.05249568027173207308618487084, 2.28577097016992627238319050224, 3.16210359405961876123728902440, 4.31136069549469982418368994431, 5.08668982455218909441288772527, 6.01668921505784330015371331683, 6.77343651103215939858800134196, 7.965626980883915701642420089055, 9.195120008723961496640752713863, 10.54467868267063436405676907601
