| L(s) = 1 | − 2-s + 3-s + 4-s + 3.25·5-s − 6-s + 3.01·7-s − 8-s + 9-s − 3.25·10-s + 0.491·11-s + 12-s + 13-s − 3.01·14-s + 3.25·15-s + 16-s + 3.62·17-s − 18-s + 5.43·19-s + 3.25·20-s + 3.01·21-s − 0.491·22-s + 2.39·23-s − 24-s + 5.61·25-s − 26-s + 27-s + 3.01·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.45·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 0.148·11-s + 0.288·12-s + 0.277·13-s − 0.805·14-s + 0.841·15-s + 0.250·16-s + 0.879·17-s − 0.235·18-s + 1.24·19-s + 0.728·20-s + 0.657·21-s − 0.104·22-s + 0.498·23-s − 0.204·24-s + 1.12·25-s − 0.196·26-s + 0.192·27-s + 0.569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.488081394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.488081394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 11 | \( 1 - 0.491T + 11T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + 0.375T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 6.35T + 71T^{2} \) |
| 73 | \( 1 - 4.00T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 - 6.02T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 4.39T + 97T^{2} \) |
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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
−7.82733784744530180631985999848, −7.40337270447726874932129590234, −6.54073404154141409778552753760, −5.65210064578791006314277777819, −5.32678904860631827969963713415, −4.29927084014417800954300623320, −3.19629846948540025439450664619, −2.48450223767255990195113548342, −1.55988886050018290359245837889, −1.17992616183597063367825765493,
1.17992616183597063367825765493, 1.55988886050018290359245837889, 2.48450223767255990195113548342, 3.19629846948540025439450664619, 4.29927084014417800954300623320, 5.32678904860631827969963713415, 5.65210064578791006314277777819, 6.54073404154141409778552753760, 7.40337270447726874932129590234, 7.82733784744530180631985999848
