| L(s) = 1 | + 3-s + 0.0349·5-s + 0.637·7-s + 9-s − 0.863·11-s − 4.36·13-s + 0.0349·15-s − 2.27·17-s − 4.02·19-s + 0.637·21-s + 8.23·23-s − 4.99·25-s + 27-s + 1.29·29-s − 1.21·31-s − 0.863·33-s + 0.0223·35-s + 7.14·37-s − 4.36·39-s + 2.42·41-s + 0.00158·43-s + 0.0349·45-s − 3.98·47-s − 6.59·49-s − 2.27·51-s + 7.67·53-s − 0.0302·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0156·5-s + 0.240·7-s + 0.333·9-s − 0.260·11-s − 1.21·13-s + 0.00903·15-s − 0.552·17-s − 0.922·19-s + 0.139·21-s + 1.71·23-s − 0.999·25-s + 0.192·27-s + 0.239·29-s − 0.217·31-s − 0.150·33-s + 0.00377·35-s + 1.17·37-s − 0.699·39-s + 0.378·41-s + 0.000241·43-s + 0.00521·45-s − 0.581·47-s − 0.941·49-s − 0.318·51-s + 1.05·53-s − 0.00407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 0.0349T + 5T^{2} \) |
| 7 | \( 1 - 0.637T + 7T^{2} \) |
| 11 | \( 1 + 0.863T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 0.00158T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.69010305312449316980640645788, −7.17503262569630470375737757660, −6.41462221651789010864291967929, −5.50946642474330710851837620110, −4.66232668471564391027938280786, −4.19924904010278338762196204265, −2.99583655317649717031099644203, −2.46627572500099346173864119336, −1.48262556560617263098500624049, 0,
1.48262556560617263098500624049, 2.46627572500099346173864119336, 2.99583655317649717031099644203, 4.19924904010278338762196204265, 4.66232668471564391027938280786, 5.50946642474330710851837620110, 6.41462221651789010864291967929, 7.17503262569630470375737757660, 7.69010305312449316980640645788
