| L(s) = 1 | − 2.65·2-s + 0.109·3-s + 5.06·4-s + 0.753·5-s − 0.292·6-s + 3.98·7-s − 8.13·8-s − 2.98·9-s − 2.00·10-s + 0.383·11-s + 0.556·12-s + 2.64·13-s − 10.5·14-s + 0.0828·15-s + 11.4·16-s + 8.14·17-s + 7.93·18-s − 5.99·19-s + 3.81·20-s + 0.438·21-s − 1.02·22-s − 1.73·23-s − 0.894·24-s − 4.43·25-s − 7.02·26-s − 0.658·27-s + 20.1·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.0634·3-s + 2.53·4-s + 0.337·5-s − 0.119·6-s + 1.50·7-s − 2.87·8-s − 0.995·9-s − 0.633·10-s + 0.115·11-s + 0.160·12-s + 0.733·13-s − 2.83·14-s + 0.0213·15-s + 2.87·16-s + 1.97·17-s + 1.87·18-s − 1.37·19-s + 0.852·20-s + 0.0956·21-s − 0.217·22-s − 0.362·23-s − 0.182·24-s − 0.886·25-s − 1.37·26-s − 0.126·27-s + 3.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 0.109T + 3T^{2} \) |
| 5 | \( 1 - 0.753T + 5T^{2} \) |
| 7 | \( 1 - 3.98T + 7T^{2} \) |
| 11 | \( 1 - 0.383T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 8.14T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 9.53T + 43T^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 4.10T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.933180842993486636053689900985, −7.48535310026595035190253044283, −6.42145951600170746805184007588, −5.84062578521542378014186966210, −5.15084151826288012512729373615, −3.77340203483779707257087560783, −2.80969564588285458756602271391, −1.77437622653453381002132735862, −1.40602793794636103038764758252, 0,
1.40602793794636103038764758252, 1.77437622653453381002132735862, 2.80969564588285458756602271391, 3.77340203483779707257087560783, 5.15084151826288012512729373615, 5.84062578521542378014186966210, 6.42145951600170746805184007588, 7.48535310026595035190253044283, 7.933180842993486636053689900985
