| L(s) = 1 | + 2-s − 3.00·3-s + 4-s − 5-s − 3.00·6-s − 4.97·7-s + 8-s + 6.02·9-s − 10-s + 3.30·11-s − 3.00·12-s − 1.23·13-s − 4.97·14-s + 3.00·15-s + 16-s + 0.120·17-s + 6.02·18-s − 0.808·19-s − 20-s + 14.9·21-s + 3.30·22-s − 2.58·23-s − 3.00·24-s + 25-s − 1.23·26-s − 9.10·27-s − 4.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 0.5·4-s − 0.447·5-s − 1.22·6-s − 1.87·7-s + 0.353·8-s + 2.00·9-s − 0.316·10-s + 0.996·11-s − 0.867·12-s − 0.342·13-s − 1.32·14-s + 0.775·15-s + 0.250·16-s + 0.0291·17-s + 1.42·18-s − 0.185·19-s − 0.223·20-s + 3.25·21-s + 0.704·22-s − 0.538·23-s − 0.613·24-s + 0.200·25-s − 0.241·26-s − 1.75·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 0.120T + 17T^{2} \) |
| 19 | \( 1 + 0.808T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 8.22T + 43T^{2} \) |
| 47 | \( 1 - 8.05T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 3.94T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 0.192T + 89T^{2} \) |
| 97 | \( 1 + 0.524T + 97T^{2} \) |
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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
−7.21316219108713315586915595057, −6.31486853810556517363592972684, −6.10133990743062433474410812894, −5.47691701910756337640826538465, −4.56044260317381771453618874627, −3.90957542008878370141402769449, −3.41286057726828707213503993903, −2.21242265543829106746702642995, −0.902410103192641493725828876630, 0,
0.902410103192641493725828876630, 2.21242265543829106746702642995, 3.41286057726828707213503993903, 3.90957542008878370141402769449, 4.56044260317381771453618874627, 5.47691701910756337640826538465, 6.10133990743062433474410812894, 6.31486853810556517363592972684, 7.21316219108713315586915595057
