| L(s) = 1 | + 0.264·2-s + 2.90·3-s − 1.92·4-s − 1.43·5-s + 0.769·6-s − 1.03·8-s + 5.46·9-s − 0.379·10-s + 5.50·11-s − 5.61·12-s + 13-s − 4.17·15-s + 3.58·16-s + 4.83·17-s + 1.44·18-s + 2.82·19-s + 2.76·20-s + 1.45·22-s − 5.99·23-s − 3.02·24-s − 2.94·25-s + 0.264·26-s + 7.16·27-s + 1.04·29-s − 1.10·30-s + 9.20·31-s + 3.02·32-s + ⋯ |
| L(s) = 1 | + 0.187·2-s + 1.67·3-s − 0.964·4-s − 0.641·5-s + 0.314·6-s − 0.367·8-s + 1.82·9-s − 0.120·10-s + 1.65·11-s − 1.62·12-s + 0.277·13-s − 1.07·15-s + 0.896·16-s + 1.17·17-s + 0.340·18-s + 0.647·19-s + 0.619·20-s + 0.310·22-s − 1.25·23-s − 0.617·24-s − 0.588·25-s + 0.0518·26-s + 1.37·27-s + 0.193·29-s − 0.201·30-s + 1.65·31-s + 0.535·32-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
| \(L(1)\) |
\(\approx\) |
\(2.294403862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294403862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.264T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 - 9.20T + 31T^{2} \) |
| 37 | \( 1 - 0.612T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 + 0.870T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 + 6.62T + 67T^{2} \) |
| 71 | \( 1 + 6.85T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 0.995T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−9.986737899983814254413615801119, −9.687521083474545194750744403811, −8.644483635497222004944499866788, −8.251911821451136091263317307404, −7.39633089165244011088360558187, −6.12089948928664736016146160193, −4.58346017908502434123911103175, −3.75545087961897204495740710977, −3.24715820190858717330829287400, −1.42185787271355655763855117191,
1.42185787271355655763855117191, 3.24715820190858717330829287400, 3.75545087961897204495740710977, 4.58346017908502434123911103175, 6.12089948928664736016146160193, 7.39633089165244011088360558187, 8.251911821451136091263317307404, 8.644483635497222004944499866788, 9.687521083474545194750744403811, 9.986737899983814254413615801119
