| L(s) = 1 | + 21.5·2-s − 48.6·4-s + 625·5-s + 2.40e3·7-s − 1.20e4·8-s + 1.34e4·10-s − 7.26e3·11-s − 1.04e5·13-s + 5.16e4·14-s − 2.34e5·16-s − 3.77e5·17-s + 6.99e5·19-s − 3.03e4·20-s − 1.56e5·22-s + 1.41e5·23-s + 3.90e5·25-s − 2.25e6·26-s − 1.16e5·28-s − 4.83e6·29-s + 9.21e6·31-s + 1.12e6·32-s − 8.13e6·34-s + 1.50e6·35-s + 1.62e7·37-s + 1.50e7·38-s − 7.54e6·40-s + 2.26e6·41-s + ⋯ |
| L(s) = 1 | + 0.951·2-s − 0.0949·4-s + 0.447·5-s + 0.377·7-s − 1.04·8-s + 0.425·10-s − 0.149·11-s − 1.01·13-s + 0.359·14-s − 0.896·16-s − 1.09·17-s + 1.23·19-s − 0.0424·20-s − 0.142·22-s + 0.105·23-s + 0.200·25-s − 0.969·26-s − 0.0358·28-s − 1.27·29-s + 1.79·31-s + 0.189·32-s − 1.04·34-s + 0.169·35-s + 1.42·37-s + 1.17·38-s − 0.465·40-s + 0.125·41-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
| \(L(5)\) |
\(\approx\) |
\(2.994901046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994901046\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 21.5T + 512T^{2} \) |
| 11 | \( 1 + 7.26e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.04e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.77e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.99e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.41e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.83e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.21e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.62e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.26e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 5.43e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.34e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.47e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.69e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.24e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.66e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.75e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.64e8T + 7.60e17T^{2} \) |
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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
−9.963000289027969521614672242423, −9.313281351139054131498164596441, −8.207677082218323343667623003106, −7.04077319652001347096592919569, −5.96281136594673635189779252019, −5.05241587647620470837437701755, −4.38930595338330187375075741352, −3.09292665285183073700157081443, −2.15971062799013188676711471893, −0.63894109696943917112414608145,
0.63894109696943917112414608145, 2.15971062799013188676711471893, 3.09292665285183073700157081443, 4.38930595338330187375075741352, 5.05241587647620470837437701755, 5.96281136594673635189779252019, 7.04077319652001347096592919569, 8.207677082218323343667623003106, 9.313281351139054131498164596441, 9.963000289027969521614672242423
