| L(s) = 1 | + 2-s − 0.148·3-s + 4-s − 1.78·5-s − 0.148·6-s − 0.469·7-s + 8-s − 2.97·9-s − 1.78·10-s − 5.56·11-s − 0.148·12-s − 6.07·13-s − 0.469·14-s + 0.264·15-s + 16-s + 6.38·17-s − 2.97·18-s − 0.00859·19-s − 1.78·20-s + 0.0696·21-s − 5.56·22-s + 23-s − 0.148·24-s − 1.81·25-s − 6.07·26-s + 0.886·27-s − 0.469·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0856·3-s + 0.5·4-s − 0.797·5-s − 0.0605·6-s − 0.177·7-s + 0.353·8-s − 0.992·9-s − 0.564·10-s − 1.67·11-s − 0.0428·12-s − 1.68·13-s − 0.125·14-s + 0.0683·15-s + 0.250·16-s + 1.54·17-s − 0.701·18-s − 0.00197·19-s − 0.398·20-s + 0.0151·21-s − 1.18·22-s + 0.208·23-s − 0.0302·24-s − 0.363·25-s − 1.19·26-s + 0.170·27-s − 0.0886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.334267377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334267377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.148T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 0.469T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 19 | \( 1 + 0.00859T + 19T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 7.76T + 41T^{2} \) |
| 43 | \( 1 - 0.302T + 43T^{2} \) |
| 47 | \( 1 + 2.07T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 7.46T + 97T^{2} \) |
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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
−7.75816865045097072256973640243, −7.64350926529822068762325488959, −6.59038662311379750436376962462, −5.76729531017395863649847628842, −4.99625335034725702201019363448, −4.78469400447327277936990178121, −3.43468073955612075331125302744, −2.96767720951768285075183764209, −2.24946034199879939676193163910, −0.50944724576768249968725835914,
0.50944724576768249968725835914, 2.24946034199879939676193163910, 2.96767720951768285075183764209, 3.43468073955612075331125302744, 4.78469400447327277936990178121, 4.99625335034725702201019363448, 5.76729531017395863649847628842, 6.59038662311379750436376962462, 7.64350926529822068762325488959, 7.75816865045097072256973640243
