| L(s) = 1 | − 0.869·2-s + 3-s − 1.24·4-s + 3.55·5-s − 0.869·6-s + 3.07·7-s + 2.82·8-s + 9-s − 3.09·10-s + 0.484·11-s − 1.24·12-s + 13-s − 2.67·14-s + 3.55·15-s + 0.0321·16-s − 6.60·17-s − 0.869·18-s − 0.185·19-s − 4.42·20-s + 3.07·21-s − 0.421·22-s − 1.09·23-s + 2.82·24-s + 7.65·25-s − 0.869·26-s + 27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | − 0.615·2-s + 0.577·3-s − 0.621·4-s + 1.59·5-s − 0.355·6-s + 1.16·7-s + 0.997·8-s + 0.333·9-s − 0.978·10-s + 0.146·11-s − 0.358·12-s + 0.277·13-s − 0.714·14-s + 0.918·15-s + 0.00802·16-s − 1.60·17-s − 0.205·18-s − 0.0426·19-s − 0.989·20-s + 0.670·21-s − 0.0899·22-s − 0.227·23-s + 0.575·24-s + 1.53·25-s − 0.170·26-s + 0.192·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417942789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417942789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.869T + 2T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 - 0.484T + 11T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 + 0.185T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 + 0.683T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−8.642543774257402909806481051534, −8.016616314646046525095457199295, −7.10621080366654141201710979274, −6.27742033452972948316552695228, −5.40434218544396587873594784747, −4.67063093444286471938258298210, −4.03536446593887909666673473794, −2.51735305347302652187583179280, −1.90462528698979966228147575871, −1.03809344199777607150505993163,
1.03809344199777607150505993163, 1.90462528698979966228147575871, 2.51735305347302652187583179280, 4.03536446593887909666673473794, 4.67063093444286471938258298210, 5.40434218544396587873594784747, 6.27742033452972948316552695228, 7.10621080366654141201710979274, 8.016616314646046525095457199295, 8.642543774257402909806481051534
