| L(s) = 1 | + 1.37·2-s + 3·3-s − 6.10·4-s + 4.12·6-s + 7·7-s − 19.4·8-s + 9·9-s + 15.2·11-s − 18.3·12-s + 76.7·13-s + 9.63·14-s + 22.1·16-s − 96.7·17-s + 12.3·18-s − 14.1·19-s + 21·21-s + 21.0·22-s − 75.7·23-s − 58.2·24-s + 105.·26-s + 27·27-s − 42.7·28-s + 89.9·29-s + 289.·31-s + 185.·32-s + 45.8·33-s − 133.·34-s + ⋯ |
| L(s) = 1 | + 0.486·2-s + 0.577·3-s − 0.763·4-s + 0.280·6-s + 0.377·7-s − 0.857·8-s + 0.333·9-s + 0.418·11-s − 0.440·12-s + 1.63·13-s + 0.183·14-s + 0.345·16-s − 1.37·17-s + 0.162·18-s − 0.171·19-s + 0.218·21-s + 0.203·22-s − 0.686·23-s − 0.495·24-s + 0.797·26-s + 0.192·27-s − 0.288·28-s + 0.576·29-s + 1.67·31-s + 1.02·32-s + 0.241·33-s − 0.671·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.755995104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.755995104\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.37T + 8T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 14.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 389.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 679.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 891.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 416.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 814.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−10.45913743713010882283085099564, −9.315126259081445383456921478838, −8.679008564840977553790709761432, −8.065209444911918777904805129094, −6.60909373085913891498968973311, −5.79103803939011231087157379419, −4.38919649214022506706923322551, −3.95760365285180002592721842104, −2.57923575051564932741480863025, −0.980134954354101282695724802418,
0.980134954354101282695724802418, 2.57923575051564932741480863025, 3.95760365285180002592721842104, 4.38919649214022506706923322551, 5.79103803939011231087157379419, 6.60909373085913891498968973311, 8.065209444911918777904805129094, 8.679008564840977553790709761432, 9.315126259081445383456921478838, 10.45913743713010882283085099564
