| L(s) = 1 | + 1.86·2-s − 3.34·3-s + 1.48·4-s + 0.866·5-s − 6.24·6-s − 0.965·8-s + 8.21·9-s + 1.61·10-s − 3.86·11-s − 4.96·12-s + 13-s − 2.90·15-s − 4.76·16-s − 3.34·17-s + 15.3·18-s − 5.38·19-s + 1.28·20-s − 7.21·22-s − 5.24·23-s + 3.23·24-s − 4.24·25-s + 1.86·26-s − 17.4·27-s + 1.69·29-s − 5.41·30-s + 7.56·31-s − 6.96·32-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.93·3-s + 0.741·4-s + 0.387·5-s − 2.55·6-s − 0.341·8-s + 2.73·9-s + 0.511·10-s − 1.16·11-s − 1.43·12-s + 0.277·13-s − 0.748·15-s − 1.19·16-s − 0.812·17-s + 3.61·18-s − 1.23·19-s + 0.287·20-s − 1.53·22-s − 1.09·23-s + 0.659·24-s − 0.849·25-s + 0.365·26-s − 3.36·27-s + 0.315·29-s − 0.988·30-s + 1.35·31-s − 1.23·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 - 0.866T + 5T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 37 | \( 1 + 4.83T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 0.215T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 - 9.03T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 + 0.418T + 89T^{2} \) |
| 97 | \( 1 + 7.11T + 97T^{2} \) |
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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
−10.53037548627621170919192519519, −9.738586663512547851786590594763, −8.192831151997316336892533342234, −6.76550769822222441770282968481, −6.22389698425433008111806872943, −5.50561498754813863688883229204, −4.76321301533434046171986553243, −4.00510973766594395050002517318, −2.17913938645567897664155777485, 0,
2.17913938645567897664155777485, 4.00510973766594395050002517318, 4.76321301533434046171986553243, 5.50561498754813863688883229204, 6.22389698425433008111806872943, 6.76550769822222441770282968481, 8.192831151997316336892533342234, 9.738586663512547851786590594763, 10.53037548627621170919192519519
