| L(s) = 1 | + 21.5·3-s + 14.7·5-s − 49·7-s + 222.·9-s − 58.5·11-s − 1.17e3·13-s + 317.·15-s + 1.49e3·17-s − 498.·19-s − 1.05e3·21-s − 1.88e3·23-s − 2.90e3·25-s − 443.·27-s − 1.91e3·29-s + 794.·31-s − 1.26e3·33-s − 721.·35-s − 2.98e3·37-s − 2.54e4·39-s + 1.19e4·41-s − 9.82e3·43-s + 3.27e3·45-s − 1.96e4·47-s + 2.40e3·49-s + 3.22e4·51-s + 1.98e4·53-s − 862.·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.263·5-s − 0.377·7-s + 0.915·9-s − 0.145·11-s − 1.93·13-s + 0.364·15-s + 1.25·17-s − 0.317·19-s − 0.523·21-s − 0.744·23-s − 0.930·25-s − 0.117·27-s − 0.422·29-s + 0.148·31-s − 0.201·33-s − 0.0995·35-s − 0.358·37-s − 2.67·39-s + 1.10·41-s − 0.809·43-s + 0.241·45-s − 1.29·47-s + 0.142·49-s + 1.73·51-s + 0.971·53-s − 0.0384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 21.5T + 243T^{2} \) |
| 5 | \( 1 - 14.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 58.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 498.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 794.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.35e4T + 8.58e9T^{2} \) |
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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
−9.738355141199675348695126356487, −9.075773747329148115527048617203, −7.83289870836369999689621333451, −7.52527451886492178208321648907, −6.11379255168038803543413897974, −4.91259117863483593148531647481, −3.65046086517572875495880511182, −2.71924234906336862898323758395, −1.85283824060961844740658856721, 0,
1.85283824060961844740658856721, 2.71924234906336862898323758395, 3.65046086517572875495880511182, 4.91259117863483593148531647481, 6.11379255168038803543413897974, 7.52527451886492178208321648907, 7.83289870836369999689621333451, 9.075773747329148115527048617203, 9.738355141199675348695126356487
