| L(s) = 1 | + 1.10·2-s + 3-s − 0.779·4-s − 3.67·5-s + 1.10·6-s − 3.07·8-s + 9-s − 4.06·10-s + 1.52·11-s − 0.779·12-s + 3.67·13-s − 3.67·15-s − 1.83·16-s − 4.16·17-s + 1.10·18-s + 5.23·19-s + 2.86·20-s + 1.68·22-s + 1.53·23-s − 3.07·24-s + 8.52·25-s + 4.06·26-s + 27-s − 8.04·29-s − 4.06·30-s + 8.26·31-s + 4.11·32-s + ⋯ |
| L(s) = 1 | + 0.781·2-s + 0.577·3-s − 0.389·4-s − 1.64·5-s + 0.450·6-s − 1.08·8-s + 0.333·9-s − 1.28·10-s + 0.458·11-s − 0.225·12-s + 1.01·13-s − 0.949·15-s − 0.457·16-s − 1.00·17-s + 0.260·18-s + 1.19·19-s + 0.641·20-s + 0.358·22-s + 0.320·23-s − 0.626·24-s + 1.70·25-s + 0.796·26-s + 0.192·27-s − 1.49·29-s − 0.741·30-s + 1.48·31-s + 0.727·32-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 | \( 1 - 1.10T + 2T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 9.52T + 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.95T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−7.85176514876347272344535598838, −7.00910357498959896074922837253, −6.35683189074657407626739566943, −5.33385086965676416970321812213, −4.57471148431419215842539944175, −3.94190207569860373134832332788, −3.51449367178187850169531799079, −2.81760178004310197483045991872, −1.26818324727627886633230621058, 0,
1.26818324727627886633230621058, 2.81760178004310197483045991872, 3.51449367178187850169531799079, 3.94190207569860373134832332788, 4.57471148431419215842539944175, 5.33385086965676416970321812213, 6.35683189074657407626739566943, 7.00910357498959896074922837253, 7.85176514876347272344535598838
