| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 36·6-s − 119.·7-s + 64·8-s + 81·9-s + 263.·11-s + 144·12-s + 851.·13-s − 478.·14-s + 256·16-s + 1.28e3·17-s + 324·18-s + 2.06e3·19-s − 1.07e3·21-s + 1.05e3·22-s + 55.5·23-s + 576·24-s + 3.40e3·26-s + 729·27-s − 1.91e3·28-s − 5.98e3·29-s + 4.78e3·31-s + 1.02e3·32-s + 2.37e3·33-s + 5.14e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.923·7-s + 0.353·8-s + 0.333·9-s + 0.657·11-s + 0.288·12-s + 1.39·13-s − 0.652·14-s + 0.250·16-s + 1.08·17-s + 0.235·18-s + 1.30·19-s − 0.533·21-s + 0.464·22-s + 0.0218·23-s + 0.204·24-s + 0.987·26-s + 0.192·27-s − 0.461·28-s − 1.32·29-s + 0.893·31-s + 0.176·32-s + 0.379·33-s + 0.763·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.831834924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.831834924\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 119.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 263.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 851.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 55.5T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.64e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.27e4T + 8.58e9T^{2} \) |
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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
−12.29643730245845403276672695634, −11.28308618070272093054267334375, −10.00066163253749145389971969994, −9.087649916802633096732975580321, −7.77180598445867241157920941929, −6.60794650232422382918690070031, −5.57018953323031855060694622970, −3.84461393245015871928285829475, −3.13009967947857664660796938954, −1.29112522766853830146684430484,
1.29112522766853830146684430484, 3.13009967947857664660796938954, 3.84461393245015871928285829475, 5.57018953323031855060694622970, 6.60794650232422382918690070031, 7.77180598445867241157920941929, 9.087649916802633096732975580321, 10.00066163253749145389971969994, 11.28308618070272093054267334375, 12.29643730245845403276672695634
