L(s) = 1 | + (0.951 + 0.309i)2-s + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (2.21 − 0.332i)5-s − 0.999·6-s + (0.907 − 1.24i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (2.20 + 0.367i)10-s + (1.74 + 1.27i)11-s + (−0.951 − 0.309i)12-s + (−0.363 + 0.118i)13-s + (1.24 − 0.907i)14-s + (−2.00 + 0.999i)15-s + (0.309 + 0.951i)16-s + (0.638 + 0.879i)17-s + ⋯ |
L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.549 + 0.178i)3-s + (0.404 + 0.293i)4-s + (0.988 − 0.148i)5-s − 0.408·6-s + (0.342 − 0.471i)7-s + (0.207 + 0.286i)8-s + (0.269 − 0.195i)9-s + (0.697 + 0.116i)10-s + (0.527 + 0.383i)11-s + (−0.274 − 0.0892i)12-s + (−0.100 + 0.0327i)13-s + (0.333 − 0.242i)14-s + (−0.516 + 0.257i)15-s + (0.0772 + 0.237i)16-s + (0.154 + 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.50592 + 0.400953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.50592 + 0.400953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.21 + 0.332i)T \) |
| 31 | \( 1 + (-5.47 - 1.01i)T \) |
good | 7 | \( 1 + (-0.907 + 1.24i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.74 - 1.27i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.363 - 0.118i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.638 - 0.879i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.32 + 4.07i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.62 + 3.61i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.31 - 4.05i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 - 3.70iT - 37T^{2} \) |
| 41 | \( 1 + (-0.587 + 1.80i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (7.37 + 2.39i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-9.01 + 2.92i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.47 - 4.78i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.08 - 3.33i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 2.20iT - 67T^{2} \) |
| 71 | \( 1 + (4.31 - 3.13i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.86 - 5.32i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.01 + 2.91i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.57 + 0.512i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.341 + 0.248i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (8.18 - 11.2i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.29040549640905476297732225778, −9.375302281079798715265468898716, −8.457191705251392019988453966136, −7.19726491403596008924897337352, −6.58928667242422207965262190536, −5.69493955378992300286334059139, −4.86676466907403245188980855524, −4.13193886585312078847274856357, −2.68861456971514494358464000001, −1.33833282134612237911271594354,
1.38036237328263060707813157390, 2.42614424017791651580609030454, 3.71326233645429822458142199230, 4.92283055478896282372035335301, 5.77576724158036729166862575973, 6.18470573386407612889714863049, 7.25963682945158242292567742603, 8.321050868685507842928681243361, 9.487382177740762111522573526095, 10.07472281640626371314771948448