| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.00 + 1.41i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (0.520 − 1.65i)6-s + (−0.0129 − 0.123i)7-s + (−0.587 + 0.809i)8-s + (−0.976 − 2.83i)9-s + (−0.669 + 0.743i)10-s + (−1.53 − 0.684i)11-s + (0.0150 + 1.73i)12-s + (0.248 + 1.17i)13-s + (0.0505 + 0.113i)14-s + (−0.166 + 1.72i)15-s + (0.309 − 0.951i)16-s + (−0.251 + 0.111i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.580 + 0.814i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.212 − 0.674i)6-s + (−0.00490 − 0.0467i)7-s + (−0.207 + 0.286i)8-s + (−0.325 − 0.945i)9-s + (−0.211 + 0.235i)10-s + (−0.463 − 0.206i)11-s + (0.00434 + 0.499i)12-s + (0.0690 + 0.324i)13-s + (0.0135 + 0.0303i)14-s + (−0.0428 + 0.445i)15-s + (0.0772 − 0.237i)16-s + (−0.0609 + 0.0271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853983 + 0.434721i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853983 + 0.434721i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.00 - 1.41i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-3.31 - 4.46i)T \) |
| good | 7 | \( 1 + (0.0129 + 0.123i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.53 + 0.684i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.248 - 1.17i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.251 - 0.111i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.89 - 1.04i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.05i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.71 + 8.36i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-8.89 - 5.13i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.07 + 3.67i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.901 + 4.24i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.81 - 0.915i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.920 - 8.75i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (0.903 - 0.813i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 3.06iT - 61T^{2} \) |
| 67 | \( 1 + (-3.27 - 5.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-15.0 - 1.58i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.31 - 5.20i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 8.64i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.69 + 4.10i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (7.19 - 5.22i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.57 + 2.59i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.01562938360719136482156434685, −9.516258678623153442781616226275, −8.723651426746099045893360037882, −7.75846893256840259649445063631, −6.72745663819121748940504981160, −5.79628702899264360059297377423, −5.16585438726998758719392382812, −4.00850143884226494225855529642, −2.70152114814167739491270254714, −0.970679223541310389793635154235,
0.830657133455405789130779012073, 2.13579083529852055569581713889, 3.13193572978757996465683409513, 4.88718724501142938882409693284, 5.76003816610087575507332186656, 6.68856658166824486563476783691, 7.44181856719610948759163619062, 8.123470772573632562725552993582, 9.185444444627031274422051317400, 9.979853182275646693821310742266
