| L(s) = 1 | + 2-s − 2.42·3-s + 4-s + 4.30·5-s − 2.42·6-s + 2.46·7-s + 8-s + 2.87·9-s + 4.30·10-s − 0.0348·11-s − 2.42·12-s + 13-s + 2.46·14-s − 10.4·15-s + 16-s − 0.619·17-s + 2.87·18-s + 4.30·20-s − 5.98·21-s − 0.0348·22-s − 2.96·23-s − 2.42·24-s + 13.5·25-s + 26-s + 0.299·27-s + 2.46·28-s + 2.42·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 1.92·5-s − 0.989·6-s + 0.933·7-s + 0.353·8-s + 0.958·9-s + 1.36·10-s − 0.0105·11-s − 0.699·12-s + 0.277·13-s + 0.660·14-s − 2.69·15-s + 0.250·16-s − 0.150·17-s + 0.677·18-s + 0.963·20-s − 1.30·21-s − 0.00742·22-s − 0.618·23-s − 0.494·24-s + 2.71·25-s + 0.196·26-s + 0.0577·27-s + 0.466·28-s + 0.450·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.656857439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656857439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 0.0348T + 11T^{2} \) |
| 17 | \( 1 + 0.619T + 17T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 - 2.42T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 + 8.34T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 6.45T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.34646012573885188404364435533, −6.60036271282703006369544009763, −6.17277900997418032849639193274, −5.60929775609652708320846894824, −5.05945800585691313956996149535, −4.71855908207520993349731908873, −3.55738404278243695650784547183, −2.35411324114176313709395399363, −1.78591976889950933767121154993, −0.931777385522273918669548453077,
0.931777385522273918669548453077, 1.78591976889950933767121154993, 2.35411324114176313709395399363, 3.55738404278243695650784547183, 4.71855908207520993349731908873, 5.05945800585691313956996149535, 5.60929775609652708320846894824, 6.17277900997418032849639193274, 6.60036271282703006369544009763, 7.34646012573885188404364435533
