| L(s) = 1 | + 10.5·2-s + 5.80·3-s + 78.8·4-s + 25·5-s + 61.1·6-s − 179.·7-s + 493.·8-s − 209.·9-s + 263.·10-s − 121·11-s + 457.·12-s − 195.·13-s − 1.89e3·14-s + 145.·15-s + 2.66e3·16-s − 1.28e3·17-s − 2.20e3·18-s + 361·19-s + 1.97e3·20-s − 1.04e3·21-s − 1.27e3·22-s + 3.48e3·23-s + 2.86e3·24-s + 625·25-s − 2.05e3·26-s − 2.62e3·27-s − 1.41e4·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.372·3-s + 2.46·4-s + 0.447·5-s + 0.692·6-s − 1.38·7-s + 2.72·8-s − 0.861·9-s + 0.832·10-s − 0.301·11-s + 0.917·12-s − 0.320·13-s − 2.57·14-s + 0.166·15-s + 2.60·16-s − 1.07·17-s − 1.60·18-s + 0.229·19-s + 1.10·20-s − 0.515·21-s − 0.561·22-s + 1.37·23-s + 1.01·24-s + 0.200·25-s − 0.596·26-s − 0.693·27-s − 3.41·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.5T + 32T^{2} \) |
| 3 | \( 1 - 5.80T + 243T^{2} \) |
| 7 | \( 1 + 179.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 195.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.28e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.78e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.55e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.21e4T + 8.58e9T^{2} \) |
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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
−8.909514453861742413203722153736, −7.46845391876771245918266598059, −6.77928870175474074073270689787, −5.98381054732405491928482514346, −5.38222768385186044796543805232, −4.36634585553309483690055079057, −3.23687164420562229801578272439, −2.89144346054580976455833919375, −1.90023477696419727356046448855, 0,
1.90023477696419727356046448855, 2.89144346054580976455833919375, 3.23687164420562229801578272439, 4.36634585553309483690055079057, 5.38222768385186044796543805232, 5.98381054732405491928482514346, 6.77928870175474074073270689787, 7.46845391876771245918266598059, 8.909514453861742413203722153736
