| L(s) = 1 | + 1.49·2-s + 3.16·3-s + 0.240·4-s − 5-s + 4.74·6-s + 7-s − 2.63·8-s + 7.04·9-s − 1.49·10-s − 1.64·11-s + 0.761·12-s − 5.51·13-s + 1.49·14-s − 3.16·15-s − 4.42·16-s − 4.91·17-s + 10.5·18-s − 3.24·19-s − 0.240·20-s + 3.16·21-s − 2.46·22-s − 5.69·23-s − 8.34·24-s + 25-s − 8.25·26-s + 12.8·27-s + 0.240·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 1.82·3-s + 0.120·4-s − 0.447·5-s + 1.93·6-s + 0.377·7-s − 0.931·8-s + 2.34·9-s − 0.473·10-s − 0.495·11-s + 0.219·12-s − 1.52·13-s + 0.400·14-s − 0.818·15-s − 1.10·16-s − 1.19·17-s + 2.48·18-s − 0.743·19-s − 0.0537·20-s + 0.691·21-s − 0.524·22-s − 1.18·23-s − 1.70·24-s + 0.200·25-s − 1.61·26-s + 2.46·27-s + 0.0454·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 - 3.16T + 3T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + 0.264T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 - 0.458T + 47T^{2} \) |
| 53 | \( 1 - 3.98T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 6.03T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−7.45305323801433117148997170530, −7.02627118869076636823257275184, −6.05192322386567555603621703305, −4.98784991234354158733457793181, −4.31712375526247392540162547164, −4.14987648978464673027490485980, −3.01277604251244377896058506638, −2.59359342876128453273450500832, −1.88866726588259826499827091103, 0,
1.88866726588259826499827091103, 2.59359342876128453273450500832, 3.01277604251244377896058506638, 4.14987648978464673027490485980, 4.31712375526247392540162547164, 4.98784991234354158733457793181, 6.05192322386567555603621703305, 7.02627118869076636823257275184, 7.45305323801433117148997170530
