| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (1.81 + 2.09i)5-s + (0.163 + 0.105i)7-s + (0.415 − 0.909i)8-s + (−2.33 + 1.49i)10-s + (0.413 + 2.87i)11-s + (0.910 − 0.585i)13-s + (−0.127 + 0.146i)14-s + (0.841 + 0.540i)16-s + (−4.45 + 1.30i)17-s + (4.24 + 1.24i)19-s + (−1.15 − 2.52i)20-s − 2.90·22-s + (4.73 + 0.757i)23-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.479 − 0.140i)4-s + (0.811 + 0.936i)5-s + (0.0617 + 0.0396i)7-s + (0.146 − 0.321i)8-s + (−0.737 + 0.473i)10-s + (0.124 + 0.866i)11-s + (0.252 − 0.162i)13-s + (−0.0339 + 0.0392i)14-s + (0.210 + 0.135i)16-s + (−1.08 + 0.317i)17-s + (0.974 + 0.286i)19-s + (−0.257 − 0.563i)20-s − 0.619·22-s + (0.987 + 0.158i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.817736 + 1.11646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817736 + 1.11646i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-4.73 - 0.757i)T \) |
| good | 5 | \( 1 + (-1.81 - 2.09i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.163 - 0.105i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.413 - 2.87i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.910 + 0.585i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.45 - 1.30i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.24 - 1.24i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (8.83 - 2.59i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.85 - 6.25i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.17 + 3.66i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.76 + 4.34i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.14 - 6.88i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 + (-3.48 - 2.23i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-9.01 + 5.79i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.19 - 2.61i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 14.4i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.81 + 12.6i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.86 + 0.841i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.15 + 1.38i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.531 - 0.613i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (2.81 + 6.16i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.09 - 9.34i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.23091801595900585081895241247, −10.50480715661418549696874401066, −9.563450648183368113310769850804, −8.873440254137897445350993991303, −7.47851952382493367895909242281, −6.88718984266025596730361931247, −5.92991876099376010663790428641, −4.94923197650926247351320341712, −3.48148844000271675878438681917, −1.95234106407029569459481523450,
1.00987733979762569761962712388, 2.42862906184095370909443622852, 3.89686901634859544997865870779, 5.09466130338135797887681113762, 5.91016600198560189280252430391, 7.32294853909673606408985849425, 8.661694478439916176845462498115, 9.140522818600518337450048159872, 9.922237075004850012529981172625, 11.20408314338793388683902662054
