| L(s) = 1 | + 2-s − 1.47·3-s + 4-s − 1.23·5-s − 1.47·6-s − 7-s + 8-s − 0.819·9-s − 1.23·10-s + 4.60·11-s − 1.47·12-s + 3.72·13-s − 14-s + 1.82·15-s + 16-s + 2.24·17-s − 0.819·18-s − 1.23·20-s + 1.47·21-s + 4.60·22-s − 6.42·23-s − 1.47·24-s − 3.47·25-s + 3.72·26-s + 5.64·27-s − 28-s − 5.87·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.852·3-s + 0.5·4-s − 0.551·5-s − 0.602·6-s − 0.377·7-s + 0.353·8-s − 0.273·9-s − 0.390·10-s + 1.38·11-s − 0.426·12-s + 1.03·13-s − 0.267·14-s + 0.470·15-s + 0.250·16-s + 0.544·17-s − 0.193·18-s − 0.275·20-s + 0.322·21-s + 0.981·22-s − 1.33·23-s − 0.301·24-s − 0.695·25-s + 0.730·26-s + 1.08·27-s − 0.188·28-s − 1.09·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922120594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922120594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 0.686T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 1.30T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6.41T + 83T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−7.989869389954899296192745607736, −7.44213807034923796015719458145, −6.42894226849414140100657387079, −5.99743672811815688487951316313, −5.59632745877102753038450918594, −4.31742436166262929803777725030, −3.93345262716821912914508772952, −3.17716763661292285382536088261, −1.86866059134825338019145539298, −0.71693699056039809624715397038,
0.71693699056039809624715397038, 1.86866059134825338019145539298, 3.17716763661292285382536088261, 3.93345262716821912914508772952, 4.31742436166262929803777725030, 5.59632745877102753038450918594, 5.99743672811815688487951316313, 6.42894226849414140100657387079, 7.44213807034923796015719458145, 7.989869389954899296192745607736
