| L(s) = 1 | − 2.17·2-s − 3-s + 2.70·4-s + 2.17·6-s + 1.17·7-s − 1.53·8-s + 9-s + 2·11-s − 2.70·12-s − 0.0917·13-s − 2.53·14-s − 2.07·16-s + 6.04·17-s − 2.17·18-s − 1.17·21-s − 4.34·22-s + 1.70·23-s + 1.53·24-s + 0.199·26-s − 27-s + 3.17·28-s + 7.95·29-s + 31-s + 7.58·32-s − 2·33-s − 13.1·34-s + 2.70·36-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.35·4-s + 0.885·6-s + 0.442·7-s − 0.544·8-s + 0.333·9-s + 0.603·11-s − 0.782·12-s − 0.0254·13-s − 0.678·14-s − 0.519·16-s + 1.46·17-s − 0.511·18-s − 0.255·21-s − 0.925·22-s + 0.356·23-s + 0.314·24-s + 0.0390·26-s − 0.192·27-s + 0.599·28-s + 1.47·29-s + 0.179·31-s + 1.34·32-s − 0.348·33-s − 2.25·34-s + 0.451·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7890274385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7890274385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.0917T + 13T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 7.95T + 29T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 4.68T + 41T^{2} \) |
| 43 | \( 1 - 0.738T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.985908184429860446485889098946, −8.288276662873675635848314837362, −7.62931128118857622962563439027, −6.90643646802457637728339664382, −6.13362597680536406123168661085, −5.14890925008326872449180317407, −4.22650392764832840118650585533, −2.90417715574532311275650740081, −1.57249068147040263824361204798, −0.814361614955195422935021510323,
0.814361614955195422935021510323, 1.57249068147040263824361204798, 2.90417715574532311275650740081, 4.22650392764832840118650585533, 5.14890925008326872449180317407, 6.13362597680536406123168661085, 6.90643646802457637728339664382, 7.62931128118857622962563439027, 8.288276662873675635848314837362, 8.985908184429860446485889098946
