| L(s) = 1 | − 2-s + 3-s + 4-s + 0.847·5-s − 6-s − 7-s − 8-s + 9-s − 0.847·10-s + 3.26·11-s + 12-s + 3.92·13-s + 14-s + 0.847·15-s + 16-s − 0.449·17-s − 18-s + 6.75·19-s + 0.847·20-s − 21-s − 3.26·22-s + 6.71·23-s − 24-s − 4.28·25-s − 3.92·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.379·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.268·10-s + 0.985·11-s + 0.288·12-s + 1.08·13-s + 0.267·14-s + 0.218·15-s + 0.250·16-s − 0.109·17-s − 0.235·18-s + 1.54·19-s + 0.189·20-s − 0.218·21-s − 0.696·22-s + 1.39·23-s − 0.204·24-s − 0.856·25-s − 0.768·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.450912064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.450912064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 0.847T + 5T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 + 0.449T + 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 - 5.53T + 37T^{2} \) |
| 41 | \( 1 + 0.406T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 - 9.43T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 - 2.22T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−7.68509621933776127531014543860, −7.43361090602533478182995703177, −6.44858601132754831859679473018, −6.03665051053071332932544055332, −5.14240038049225949352186376669, −4.00842184597131008024271877746, −3.40881042540234213098554956156, −2.63893099568619463629676189073, −1.56990815607439274198368494897, −0.922194821246509734207474709875,
0.922194821246509734207474709875, 1.56990815607439274198368494897, 2.63893099568619463629676189073, 3.40881042540234213098554956156, 4.00842184597131008024271877746, 5.14240038049225949352186376669, 6.03665051053071332932544055332, 6.44858601132754831859679473018, 7.43361090602533478182995703177, 7.68509621933776127531014543860
