| L(s) = 1 | + 5.29·2-s + 0.842·3-s + 20.0·4-s + 17.1·5-s + 4.46·6-s + 29.9·7-s + 63.8·8-s − 26.2·9-s + 90.7·10-s + 16.8·12-s − 13·13-s + 158.·14-s + 14.4·15-s + 177.·16-s − 83.3·17-s − 139.·18-s − 30.7·19-s + 343.·20-s + 25.2·21-s − 15.4·23-s + 53.7·24-s + 168.·25-s − 68.8·26-s − 44.8·27-s + 600.·28-s + 145.·29-s + 76.4·30-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.162·3-s + 2.50·4-s + 1.53·5-s + 0.303·6-s + 1.61·7-s + 2.82·8-s − 0.973·9-s + 2.86·10-s + 0.406·12-s − 0.277·13-s + 3.02·14-s + 0.248·15-s + 2.77·16-s − 1.18·17-s − 1.82·18-s − 0.371·19-s + 3.84·20-s + 0.261·21-s − 0.139·23-s + 0.457·24-s + 1.34·25-s − 0.519·26-s − 0.319·27-s + 4.05·28-s + 0.929·29-s + 0.464·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.06857077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.06857077\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 5.29T + 8T^{2} \) |
| 3 | \( 1 - 0.842T + 27T^{2} \) |
| 5 | \( 1 - 17.1T + 125T^{2} \) |
| 7 | \( 1 - 29.9T + 343T^{2} \) |
| 17 | \( 1 + 83.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 94.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 403.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 122.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 326.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 764.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 645.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 349.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−8.914658110892383388714430565653, −8.193145110033701522658718840985, −7.06262296801802737843936740118, −6.21012567111852565332879734692, −5.66361722592278394310343738449, −4.86434244506583083633777215130, −4.38952137119283432715725872567, −2.87366515688262114030534728291, −2.27242812527076461734948166093, −1.52863750353103900074758917292,
1.52863750353103900074758917292, 2.27242812527076461734948166093, 2.87366515688262114030534728291, 4.38952137119283432715725872567, 4.86434244506583083633777215130, 5.66361722592278394310343738449, 6.21012567111852565332879734692, 7.06262296801802737843936740118, 8.193145110033701522658718840985, 8.914658110892383388714430565653
