| L(s) = 1 | + 3-s − 2·5-s + 4·7-s − 2·9-s + 2·11-s − 7·13-s − 2·15-s + 4·17-s + 6·19-s + 4·21-s − 23-s − 25-s − 5·27-s − 5·29-s + 3·31-s + 2·33-s − 8·35-s + 2·37-s − 7·39-s + 8·43-s + 4·45-s + 47-s + 9·49-s + 4·51-s + 6·53-s − 4·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s − 2/3·9-s + 0.603·11-s − 1.94·13-s − 0.516·15-s + 0.970·17-s + 1.37·19-s + 0.872·21-s − 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s + 0.538·31-s + 0.348·33-s − 1.35·35-s + 0.328·37-s − 1.12·39-s + 1.21·43-s + 0.596·45-s + 0.145·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.64422728706555, −12.35965586792210, −11.77322763073311, −11.55477192482865, −11.43997430743609, −10.57754198470127, −10.16458745952809, −9.529095967441412, −9.196845202146004, −8.772396230032393, −8.027596444436514, −7.714998825303971, −7.585045825655246, −7.226849331088579, −6.295013556193747, −5.643192468787734, −5.299890868488704, −4.731790599821310, −4.368597771242286, −3.668656285754481, −3.292666203884509, −2.567371610059048, −2.189698793806063, −1.447740073094309, −0.8158310682135202, 0,
0.8158310682135202, 1.447740073094309, 2.189698793806063, 2.567371610059048, 3.292666203884509, 3.668656285754481, 4.368597771242286, 4.731790599821310, 5.299890868488704, 5.643192468787734, 6.295013556193747, 7.226849331088579, 7.585045825655246, 7.714998825303971, 8.027596444436514, 8.772396230032393, 9.196845202146004, 9.529095967441412, 10.16458745952809, 10.57754198470127, 11.43997430743609, 11.55477192482865, 11.77322763073311, 12.35965586792210, 12.64422728706555
