| L(s) = 1 | + 20.9·2-s + 81·3-s − 72.2·4-s + 1.69e3·6-s + 3.57e3·7-s − 1.22e4·8-s + 6.56e3·9-s − 7.45e4·11-s − 5.85e3·12-s − 3.44e4·13-s + 7.49e4·14-s − 2.19e5·16-s + 3.72e5·17-s + 1.37e5·18-s − 8.35e5·19-s + 2.89e5·21-s − 1.56e6·22-s + 3.18e4·23-s − 9.92e5·24-s − 7.23e5·26-s + 5.31e5·27-s − 2.58e5·28-s − 6.73e6·29-s − 4.04e6·31-s + 1.66e6·32-s − 6.03e6·33-s + 7.81e6·34-s + ⋯ |
| L(s) = 1 | + 0.926·2-s + 0.577·3-s − 0.141·4-s + 0.535·6-s + 0.562·7-s − 1.05·8-s + 0.333·9-s − 1.53·11-s − 0.0814·12-s − 0.334·13-s + 0.521·14-s − 0.838·16-s + 1.08·17-s + 0.308·18-s − 1.47·19-s + 0.324·21-s − 1.42·22-s + 0.0237·23-s − 0.610·24-s − 0.310·26-s + 0.192·27-s − 0.0794·28-s − 1.76·29-s − 0.786·31-s + 0.280·32-s − 0.885·33-s + 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 20.9T + 512T^{2} \) |
| 7 | \( 1 - 3.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.44e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.72e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.18e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.52e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.16e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.77e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.06e9T + 7.60e17T^{2} \) |
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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
−12.67532978211908182564880865016, −11.13672719988989548844948329460, −9.861997217206490731314512137499, −8.553286756614399690630129526038, −7.54233768811465345101971763819, −5.72542528027685596045160673313, −4.74067576741076782942105997607, −3.45017035636367489990902106310, −2.15669396835982072154067108896, 0,
2.15669396835982072154067108896, 3.45017035636367489990902106310, 4.74067576741076782942105997607, 5.72542528027685596045160673313, 7.54233768811465345101971763819, 8.553286756614399690630129526038, 9.861997217206490731314512137499, 11.13672719988989548844948329460, 12.67532978211908182564880865016
