| L(s) = 1 | + 0.219·2-s − 1.95·4-s + 5-s − 0.332·7-s − 0.868·8-s + 0.219·10-s + 5.37·11-s − 0.0729·14-s + 3.71·16-s + 5.06·17-s + 2.26·19-s − 1.95·20-s + 1.18·22-s + 2.83·23-s + 25-s + 0.648·28-s + 2.90·29-s + 5.46·31-s + 2.55·32-s + 1.11·34-s − 0.332·35-s + 5.97·37-s + 0.498·38-s − 0.868·40-s − 3.73·41-s − 5.06·43-s − 10.4·44-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.975·4-s + 0.447·5-s − 0.125·7-s − 0.306·8-s + 0.0694·10-s + 1.61·11-s − 0.0195·14-s + 0.928·16-s + 1.22·17-s + 0.520·19-s − 0.436·20-s + 0.251·22-s + 0.592·23-s + 0.200·25-s + 0.122·28-s + 0.539·29-s + 0.981·31-s + 0.451·32-s + 0.190·34-s − 0.0561·35-s + 0.981·37-s + 0.0808·38-s − 0.137·40-s − 0.582·41-s − 0.772·43-s − 1.58·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.353599177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.353599177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.219T + 2T^{2} \) |
| 7 | \( 1 + 0.332T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 5.06T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−8.105786344488188113508304512051, −7.04939199783332093064007320743, −6.43278012809285356842520443278, −5.74475316094223701652324766066, −5.02181359694962127006887578588, −4.36525307011471871736990959294, −3.53327570218927946350942303765, −2.97423489740894986468448335840, −1.51316113151363091522723459848, −0.847599902754695506243994549040,
0.847599902754695506243994549040, 1.51316113151363091522723459848, 2.97423489740894986468448335840, 3.53327570218927946350942303765, 4.36525307011471871736990959294, 5.02181359694962127006887578588, 5.74475316094223701652324766066, 6.43278012809285356842520443278, 7.04939199783332093064007320743, 8.105786344488188113508304512051
