| L(s) = 1 | − 24.5·3-s − 25·5-s + 359.·9-s + 90.2·11-s + 14.4·13-s + 613.·15-s − 407.·17-s − 2.28e3·19-s − 505.·23-s + 625·25-s − 2.85e3·27-s − 3.16e3·29-s + 6.23e3·31-s − 2.21e3·33-s + 5.38e3·37-s − 354.·39-s − 1.17e4·41-s − 5.82e3·43-s − 8.98e3·45-s − 7.34e3·47-s + 9.99e3·51-s + 1.49e4·53-s − 2.25e3·55-s + 5.61e4·57-s + 4.71e4·59-s + 4.28e3·61-s − 361.·65-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s + 1.47·9-s + 0.224·11-s + 0.0236·13-s + 0.704·15-s − 0.341·17-s − 1.45·19-s − 0.199·23-s + 0.200·25-s − 0.754·27-s − 0.698·29-s + 1.16·31-s − 0.354·33-s + 0.647·37-s − 0.0373·39-s − 1.09·41-s − 0.480·43-s − 0.661·45-s − 0.485·47-s + 0.538·51-s + 0.732·53-s − 0.100·55-s + 2.29·57-s + 1.76·59-s + 0.147·61-s − 0.0105·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 24.5T + 243T^{2} \) |
| 11 | \( 1 - 90.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 14.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + 407.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 505.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.28e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.59e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e4T + 8.58e9T^{2} \) |
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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
−8.809326915426867408696309224925, −7.957075086101113429886766829394, −6.77052860418021295824087497195, −6.40271804007818659346966163826, −5.38067458857025593860079439259, −4.58909784930268988900979903297, −3.76102399815077905283025054422, −2.16968488520114866900650044002, −0.864384388988484714526849933383, 0,
0.864384388988484714526849933383, 2.16968488520114866900650044002, 3.76102399815077905283025054422, 4.58909784930268988900979903297, 5.38067458857025593860079439259, 6.40271804007818659346966163826, 6.77052860418021295824087497195, 7.957075086101113429886766829394, 8.809326915426867408696309224925
