| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s − 7-s − 3·8-s + 9-s + 2·10-s − 2·11-s − 12-s + 2·13-s − 14-s + 2·15-s − 16-s + 8·17-s + 18-s − 2·20-s − 21-s − 2·22-s − 3·24-s − 25-s + 2·26-s + 27-s + 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 1.94·17-s + 0.235·18-s − 0.447·20-s − 0.218·21-s − 0.426·22-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.899876877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.899876877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.11862548457551, −13.86713130543253, −13.46083934266190, −12.96658772526927, −12.39588082181484, −12.15015287121270, −11.40484447649278, −10.57427719885956, −10.04280172067552, −9.756363037583774, −9.348660441984495, −8.572031605900709, −8.145187294387182, −7.739836813640858, −6.776128794059372, −6.295192819614281, −5.704223432486285, −5.348402707296323, −4.687012526642912, −4.018490773904981, −3.403405152205562, −2.884191592209992, −2.397462619591363, −1.334968672305241, −0.7019524465190857,
0.7019524465190857, 1.334968672305241, 2.397462619591363, 2.884191592209992, 3.403405152205562, 4.018490773904981, 4.687012526642912, 5.348402707296323, 5.704223432486285, 6.295192819614281, 6.776128794059372, 7.739836813640858, 8.145187294387182, 8.572031605900709, 9.348660441984495, 9.756363037583774, 10.04280172067552, 10.57427719885956, 11.40484447649278, 12.15015287121270, 12.39588082181484, 12.96658772526927, 13.46083934266190, 13.86713130543253, 14.11862548457551
