| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.104 + 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−1.89 − 0.403i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)10-s + (0.662 − 0.735i)11-s + (−0.913 + 0.406i)12-s + (3.59 + 1.59i)13-s + (1.29 + 1.44i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.32 + 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.0603 + 0.574i)3-s + (0.154 + 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (−0.716 − 0.152i)7-s + (0.109 − 0.336i)8-s + (−0.326 + 0.0693i)9-s + (0.0330 − 0.314i)10-s + (0.199 − 0.221i)11-s + (−0.263 + 0.117i)12-s + (0.995 + 0.443i)13-s + (0.346 + 0.384i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.321 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0687 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0687 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.643832 + 0.689742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643832 + 0.689742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.00 - 2.44i)T \) |
| good | 7 | \( 1 + (1.89 + 0.403i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.662 + 0.735i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.59 - 1.59i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 1.47i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.68 - 1.64i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.256 - 0.790i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.80 - 4.21i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (5.84 - 10.1i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.392 + 3.73i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (9.85 - 4.38i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (8.50 - 6.17i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.55 + 0.543i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.329 + 3.13i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 3.14T + 61T^{2} \) |
| 67 | \( 1 + (4.08 + 7.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.87 + 1.88i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (7.84 - 8.71i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-3.77 - 4.19i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.434 - 4.13i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.05 - 12.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.42 - 16.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.21990296412154461097241984144, −9.659995583274900543517881236958, −8.644938716264572055857539128371, −8.169430242099284823048364928774, −6.62872427086996815105725968966, −6.35295450988356225401243225460, −4.86423047760376562147029249544, −3.66010489420469353898392582401, −3.04896120209559177821608249046, −1.50797016129001716537938730331,
0.54945519166628747279512198082, 1.96610024530698188006084913187, 3.26823724709244923257451425805, 4.68413609725926644986595798306, 5.89753044532430445459362977857, 6.41678608815714651753043086036, 7.29262285885627148620573706362, 8.369782213688314201184389918706, 8.763381916287491752379939884205, 9.789259620253987225842313594989
