| L(s) = 1 | + (0.927 + 1.23i)2-s + (0.547 − 1.46i)3-s + (−0.110 + 0.377i)4-s + (−1.20 − 1.88i)5-s + (2.32 − 0.683i)6-s + (0.876 + 4.02i)7-s + (2.32 − 0.868i)8-s + (0.409 + 0.354i)9-s + (1.21 − 3.23i)10-s + (−4.51 + 0.649i)11-s + (0.493 + 0.369i)12-s + (−3.88 − 0.844i)13-s + (−4.17 + 4.82i)14-s + (−3.42 + 0.737i)15-s + (3.89 + 2.50i)16-s + (−0.258 − 0.141i)17-s + ⋯ |
| L(s) = 1 | + (0.655 + 0.875i)2-s + (0.316 − 0.848i)3-s + (−0.0553 + 0.188i)4-s + (−0.538 − 0.842i)5-s + (0.950 − 0.278i)6-s + (0.331 + 1.52i)7-s + (0.823 − 0.307i)8-s + (0.136 + 0.118i)9-s + (0.384 − 1.02i)10-s + (−1.36 + 0.195i)11-s + (0.142 + 0.106i)12-s + (−1.07 − 0.234i)13-s + (−1.11 + 1.28i)14-s + (−0.885 + 0.190i)15-s + (0.974 + 0.626i)16-s + (−0.0627 − 0.0342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46954 + 0.151513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46954 + 0.151513i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.20 + 1.88i)T \) |
| 23 | \( 1 + (-4.78 + 0.292i)T \) |
| good | 2 | \( 1 + (-0.927 - 1.23i)T + (-0.563 + 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.547 + 1.46i)T + (-2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.876 - 4.02i)T + (-6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (4.51 - 0.649i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.88 + 0.844i)T + (11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (0.258 + 0.141i)T + (9.19 + 14.3i)T^{2} \) |
| 19 | \( 1 + (3.49 + 1.02i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.403 + 1.37i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.77 - 6.08i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.08 + 0.220i)T + (36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (6.08 + 7.02i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.29 - 0.482i)T + (32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (-9.04 + 9.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.847 + 0.184i)T + (48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (-2.33 - 3.63i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (3.31 + 1.51i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.361 + 0.270i)T + (18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (0.537 - 3.73i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.09 + 3.83i)T + (-39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (-8.40 + 5.39i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.0672 - 0.939i)T + (-82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (1.99 + 4.35i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (0.556 - 7.77i)T + (-96.0 - 13.8i)T^{2} \) |
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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
−13.47595653617069970416178161811, −12.74695145500038594325642946478, −12.15430650268759570959572535519, −10.51903942113212703148008065498, −8.844120927158842820763767544698, −7.912609627337863155271811082896, −7.07159087018403335796094142124, −5.43852214381750620195854122317, −4.85279530130069725164640120756, −2.25663827674329577445042363647,
2.79894255045760899754706089417, 3.96005912052194179558464592842, 4.73276522484044822213854635006, 7.14728381871235134513745946768, 7.913214480977011163604099152738, 9.898667129523481207521841910403, 10.66216213412910389857464052029, 11.14058673293727894219751046068, 12.56085058267010536984351143112, 13.48382024683726705956749924924
