L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.758 + 2.83i)3-s + (−0.499 − 0.866i)4-s + (−1.61 − 1.55i)5-s + (−2.07 − 2.07i)6-s + (0.455 − 1.69i)7-s + 0.999·8-s + (−4.84 − 2.79i)9-s + (2.14 − 0.619i)10-s − 5.29i·11-s + (2.83 − 0.758i)12-s + (−0.197 − 0.342i)13-s + (1.24 + 1.24i)14-s + (5.61 − 3.38i)15-s + (−0.5 + 0.866i)16-s + (3.21 + 1.85i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.438 + 1.63i)3-s + (−0.249 − 0.433i)4-s + (−0.720 − 0.693i)5-s + (−0.846 − 0.846i)6-s + (0.172 − 0.641i)7-s + 0.353·8-s + (−1.61 − 0.932i)9-s + (0.679 − 0.195i)10-s − 1.59i·11-s + (0.817 − 0.219i)12-s + (−0.0549 − 0.0951i)13-s + (0.332 + 0.332i)14-s + (1.44 − 0.873i)15-s + (−0.125 + 0.216i)16-s + (0.778 + 0.449i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.413794 - 0.144520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.413794 - 0.144520i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (1.61 + 1.55i)T \) |
| 37 | \( 1 + (-2.51 + 5.53i)T \) |
good | 3 | \( 1 + (0.758 - 2.83i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.455 + 1.69i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (0.197 + 0.342i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.21 - 1.85i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.24 + 1.40i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 7.28T + 23T^{2} \) |
| 29 | \( 1 + (1.33 + 1.33i)T + 29iT^{2} \) |
| 31 | \( 1 + (-3.75 + 3.75i)T - 31iT^{2} \) |
| 41 | \( 1 + (5.24 - 3.03i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 + (7.10 + 7.10i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.33 + 8.69i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.49 + 5.59i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (0.793 + 0.212i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (3.42 + 0.917i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.22 + 2.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.76 - 7.76i)T + 73iT^{2} \) |
| 79 | \( 1 + (12.4 + 3.33i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (0.519 + 1.93i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.441 + 0.118i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−11.04006297792805229820413319053, −10.37007662897974730322557570390, −9.467294671685231420836222058067, −8.480486517548494220648529488909, −7.944034274210518567549897978775, −6.21507407166420386614709285147, −5.40321827890047302197063494804, −4.29068644236957626114046029823, −3.67736805921886314042450089189, −0.35028194899048106830667011841,
1.72285252141415801453902056257, 2.69225250759224730327879523127, 4.43157388858496600655210564584, 6.00310994455906728578054085834, 6.98801626431232734999667799679, 7.69742751659189513025083334383, 8.409271052860105224079369400775, 9.846381720196599758995987637397, 10.78740239406687491136191841443, 11.94940695273580789068744613327