| L(s) = 1 | + 0.196·2-s + 3-s − 1.96·4-s − 1.00·5-s + 0.196·6-s − 0.778·8-s + 9-s − 0.197·10-s − 4.08·11-s − 1.96·12-s + 1.13·13-s − 1.00·15-s + 3.76·16-s + 2.29·17-s + 0.196·18-s + 4.14·19-s + 1.97·20-s − 0.803·22-s − 5.73·23-s − 0.778·24-s − 3.98·25-s + 0.222·26-s + 27-s − 0.264·29-s − 0.197·30-s + 4.67·31-s + 2.29·32-s + ⋯ |
| L(s) = 1 | + 0.138·2-s + 0.577·3-s − 0.980·4-s − 0.450·5-s + 0.0802·6-s − 0.275·8-s + 0.333·9-s − 0.0625·10-s − 1.23·11-s − 0.566·12-s + 0.313·13-s − 0.259·15-s + 0.942·16-s + 0.556·17-s + 0.0463·18-s + 0.952·19-s + 0.441·20-s − 0.171·22-s − 1.19·23-s − 0.158·24-s − 0.797·25-s + 0.0436·26-s + 0.192·27-s − 0.0490·29-s − 0.0361·30-s + 0.839·31-s + 0.406·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.196T + 2T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 1.13T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 0.264T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 - 8.01T + 37T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 + 0.631T + 79T^{2} \) |
| 83 | \( 1 + 1.68T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
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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
−7.85994623692388034720388089571, −7.41318091256426485960797435625, −6.03871378033578548392459452404, −5.60037008820032064707064782339, −4.62773100796828407433588254603, −4.10408607898154206090545989070, −3.26575823314964949594707549156, −2.57959100783407187096206284622, −1.21900499128417738753456469475, 0,
1.21900499128417738753456469475, 2.57959100783407187096206284622, 3.26575823314964949594707549156, 4.10408607898154206090545989070, 4.62773100796828407433588254603, 5.60037008820032064707064782339, 6.03871378033578548392459452404, 7.41318091256426485960797435625, 7.85994623692388034720388089571
