| L(s) = 1 | + 2-s + 2.97·3-s + 4-s − 1.77·5-s + 2.97·6-s + 7-s + 8-s + 5.85·9-s − 1.77·10-s − 1.39·11-s + 2.97·12-s + 1.35·13-s + 14-s − 5.27·15-s + 16-s + 6.33·17-s + 5.85·18-s + 6.19·19-s − 1.77·20-s + 2.97·21-s − 1.39·22-s − 2.46·23-s + 2.97·24-s − 1.85·25-s + 1.35·26-s + 8.50·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s − 0.792·5-s + 1.21·6-s + 0.377·7-s + 0.353·8-s + 1.95·9-s − 0.560·10-s − 0.420·11-s + 0.859·12-s + 0.376·13-s + 0.267·14-s − 1.36·15-s + 0.250·16-s + 1.53·17-s + 1.38·18-s + 1.42·19-s − 0.396·20-s + 0.649·21-s − 0.297·22-s − 0.513·23-s + 0.607·24-s − 0.371·25-s + 0.266·26-s + 1.63·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.083234270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.083234270\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 6.33T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 9.18T + 37T^{2} \) |
| 41 | \( 1 + 0.432T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 - 0.395T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 - 8.46T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.11T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−8.005587236611869733805733867349, −7.47531704255520513173332917005, −7.07056327170088752337609396786, −5.69089742796879861312040657901, −5.16111373476549669221207221938, −3.96137775451740736468240344320, −3.70445736343630975832999333723, −3.01306606371878539769521163357, −2.15336283471824006972780629830, −1.16738999578878228259520937618,
1.16738999578878228259520937618, 2.15336283471824006972780629830, 3.01306606371878539769521163357, 3.70445736343630975832999333723, 3.96137775451740736468240344320, 5.16111373476549669221207221938, 5.69089742796879861312040657901, 7.07056327170088752337609396786, 7.47531704255520513173332917005, 8.005587236611869733805733867349
