| L(s) = 1 | + 8.53·2-s + 28.4·3-s + 40.7·4-s − 25·5-s + 242.·6-s + 183.·7-s + 75.0·8-s + 564.·9-s − 213.·10-s − 121·11-s + 1.15e3·12-s − 548.·13-s + 1.56e3·14-s − 710.·15-s − 665.·16-s + 972.·17-s + 4.81e3·18-s + 361·19-s − 1.01e3·20-s + 5.22e3·21-s − 1.03e3·22-s + 2.15e3·23-s + 2.13e3·24-s + 625·25-s − 4.67e3·26-s + 9.13e3·27-s + 7.49e3·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.82·3-s + 1.27·4-s − 0.447·5-s + 2.74·6-s + 1.41·7-s + 0.414·8-s + 2.32·9-s − 0.674·10-s − 0.301·11-s + 2.32·12-s − 0.899·13-s + 2.13·14-s − 0.815·15-s − 0.649·16-s + 0.816·17-s + 3.50·18-s + 0.229·19-s − 0.570·20-s + 2.58·21-s − 0.454·22-s + 0.847·23-s + 0.755·24-s + 0.200·25-s − 1.35·26-s + 2.41·27-s + 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(12.74377239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.74377239\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 8.53T + 32T^{2} \) |
| 3 | \( 1 - 28.4T + 243T^{2} \) |
| 7 | \( 1 - 183.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 548.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 972.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 865.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.54e5T + 8.58e9T^{2} \) |
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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
−9.023124549372123784917778211974, −8.068524497410231838695977970762, −7.70014432226001860716362459815, −6.79628991517543449466709322520, −5.28962908354758296070350105137, −4.62945246320615609716821983751, −3.94938439001056930331954914495, −2.93532896949571199610357195048, −2.42199097352111786580488250218, −1.24867994421228399222139021251,
1.24867994421228399222139021251, 2.42199097352111786580488250218, 2.93532896949571199610357195048, 3.94938439001056930331954914495, 4.62945246320615609716821983751, 5.28962908354758296070350105137, 6.79628991517543449466709322520, 7.70014432226001860716362459815, 8.068524497410231838695977970762, 9.023124549372123784917778211974
