| L(s) = 1 | − 1.87·2-s + 1.73·3-s + 1.52·4-s − 2.87·5-s − 3.25·6-s − 0.555·7-s + 0.893·8-s + 0.0103·9-s + 5.39·10-s + 4.53·11-s + 2.64·12-s + 0.0502·13-s + 1.04·14-s − 4.98·15-s − 4.72·16-s − 5.51·17-s − 0.0194·18-s + 2.23·19-s − 4.37·20-s − 0.964·21-s − 8.51·22-s + 4.63·23-s + 1.55·24-s + 3.24·25-s − 0.0943·26-s − 5.18·27-s − 0.847·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 1.00·3-s + 0.761·4-s − 1.28·5-s − 1.32·6-s − 0.210·7-s + 0.315·8-s + 0.00344·9-s + 1.70·10-s + 1.36·11-s + 0.763·12-s + 0.0139·13-s + 0.278·14-s − 1.28·15-s − 1.18·16-s − 1.33·17-s − 0.00457·18-s + 0.512·19-s − 0.978·20-s − 0.210·21-s − 1.81·22-s + 0.967·23-s + 0.316·24-s + 0.649·25-s − 0.0185·26-s − 0.998·27-s − 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 0.555T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 0.0502T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 7.81T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 3.53T + 43T^{2} \) |
| 47 | \( 1 - 5.92T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 9.84T + 71T^{2} \) |
| 73 | \( 1 - 5.05T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 0.708T + 97T^{2} \) |
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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
−8.338983252571681393037213524016, −7.50767668332421109120870244411, −7.10330410226671805811297474369, −6.28698704755858726910321835885, −4.79707910283898939426111486213, −4.02425918955618374641732334494, −3.36291352077704810577936133741, −2.32672972918393829979983714504, −1.20500161931864769752759259402, 0,
1.20500161931864769752759259402, 2.32672972918393829979983714504, 3.36291352077704810577936133741, 4.02425918955618374641732334494, 4.79707910283898939426111486213, 6.28698704755858726910321835885, 7.10330410226671805811297474369, 7.50767668332421109120870244411, 8.338983252571681393037213524016
