| L(s) = 1 | + (0.540 + 0.841i)2-s + (1.59 + 0.229i)3-s + (−0.415 + 0.909i)4-s + (1.85 + 1.25i)5-s + (0.671 + 1.46i)6-s + (1.35 − 1.17i)7-s + (−0.989 + 0.142i)8-s + (−0.374 − 0.109i)9-s + (−0.0492 + 2.23i)10-s + (−4.81 − 3.09i)11-s + (−0.873 + 1.35i)12-s + (−1.68 − 1.45i)13-s + (1.71 + 0.504i)14-s + (2.67 + 2.42i)15-s + (−0.654 − 0.755i)16-s + (3.28 − 1.50i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (0.923 + 0.132i)3-s + (−0.207 + 0.454i)4-s + (0.829 + 0.559i)5-s + (0.273 + 0.599i)6-s + (0.511 − 0.442i)7-s + (−0.349 + 0.0503i)8-s + (−0.124 − 0.0366i)9-s + (−0.0155 + 0.706i)10-s + (−1.45 − 0.932i)11-s + (−0.252 + 0.392i)12-s + (−0.466 − 0.404i)13-s + (0.458 + 0.134i)14-s + (0.691 + 0.626i)15-s + (−0.163 − 0.188i)16-s + (0.797 − 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74392 + 0.945069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74392 + 0.945069i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 5 | \( 1 + (-1.85 - 1.25i)T \) |
| 23 | \( 1 + (-4.17 - 2.36i)T \) |
| good | 3 | \( 1 + (-1.59 - 0.229i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 1.17i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.81 + 3.09i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.68 + 1.45i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.28 + 1.50i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.15 - 6.90i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.955 + 2.09i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.29 + 9.02i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.17 - 3.99i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.64 + 1.65i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (2.25 + 0.324i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 3.81iT - 47T^{2} \) |
| 53 | \( 1 + (2.96 - 2.57i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.34 + 7.32i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.721 + 5.01i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (2.00 + 3.12i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.655 - 0.421i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-13.8 - 6.32i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.94 - 8.01i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.32 + 14.7i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.0331 + 0.230i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.29 - 11.2i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−12.82079170043456558429045368058, −11.28253262056447146444864867169, −10.31278147831994916517841406228, −9.383538654316940271446927152304, −8.016586387296919327895412407716, −7.72751383328858558040311692335, −6.06145427700013225502587367741, −5.27557917833067353330490106686, −3.54908261866501904133442841800, −2.52656569821969977553944097905,
2.01677731271154320299033765550, 2.79304035085044906587922757191, 4.76616650758760275888003617500, 5.39469664640552723096385702332, 7.11430352665904941518898042257, 8.438903694575277622056334118530, 9.080161873197383316529538868030, 10.11818471238221578100440874498, 11.03538667547110114907843060322, 12.43214796780550162239363268276
