| L(s) = 1 | − 2-s − 1.30·3-s + 4-s + 1.77·5-s + 1.30·6-s − 3.58·7-s − 8-s − 1.28·9-s − 1.77·10-s + 5.72·11-s − 1.30·12-s − 6.13·13-s + 3.58·14-s − 2.32·15-s + 16-s + 0.292·17-s + 1.28·18-s + 19-s + 1.77·20-s + 4.69·21-s − 5.72·22-s − 2.15·23-s + 1.30·24-s − 1.84·25-s + 6.13·26-s + 5.61·27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.756·3-s + 0.5·4-s + 0.794·5-s + 0.534·6-s − 1.35·7-s − 0.353·8-s − 0.428·9-s − 0.561·10-s + 1.72·11-s − 0.378·12-s − 1.70·13-s + 0.958·14-s − 0.601·15-s + 0.250·16-s + 0.0708·17-s + 0.302·18-s + 0.229·19-s + 0.397·20-s + 1.02·21-s − 1.22·22-s − 0.449·23-s + 0.267·24-s − 0.368·25-s + 1.20·26-s + 1.07·27-s − 0.677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 3.47T + 37T^{2} \) |
| 41 | \( 1 + 0.0748T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 + 9.24T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 7.77T + 59T^{2} \) |
| 61 | \( 1 - 0.142T + 61T^{2} \) |
| 67 | \( 1 + 0.163T + 67T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 - 6.03T + 97T^{2} \) |
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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
−7.28426150253197789866782516123, −6.53767235436546320465166663310, −6.41295508965685672122328273917, −5.63847155241630685788596768550, −4.85933937405868047680054050129, −3.80936031388137153770175778050, −2.90179104010472676739982166186, −2.16704122374191278345330818679, −0.996548075999839258116444221998, 0,
0.996548075999839258116444221998, 2.16704122374191278345330818679, 2.90179104010472676739982166186, 3.80936031388137153770175778050, 4.85933937405868047680054050129, 5.63847155241630685788596768550, 6.41295508965685672122328273917, 6.53767235436546320465166663310, 7.28426150253197789866782516123
