| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.766 + 0.642i)6-s + (0.673 + 1.16i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.173 − 0.984i)10-s + (1.43 − 2.49i)11-s + (0.499 + 0.866i)12-s + (4.14 − 3.47i)13-s + (1.26 − 0.460i)14-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.0393 + 0.223i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (0.420 − 0.152i)5-s + (−0.312 + 0.262i)6-s + (0.254 + 0.441i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.0549 − 0.311i)10-s + (0.434 − 0.751i)11-s + (0.144 + 0.249i)12-s + (1.14 − 0.964i)13-s + (0.338 − 0.123i)14-s + (−0.242 − 0.0883i)15-s + (0.191 + 0.160i)16-s + (−0.00954 + 0.0541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.737645 - 1.17071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737645 - 1.17071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (1.15 + 4.20i)T \) |
| good | 7 | \( 1 + (-0.673 - 1.16i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.43 + 2.49i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.14 + 3.47i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0393 - 0.223i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (5.79 + 2.11i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.21 + 6.87i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.81 - 6.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.879T + 37T^{2} \) |
| 41 | \( 1 + (-4.28 - 3.59i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.91 - 2.15i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.376 + 2.13i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (2.76 + 1.00i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.03 + 5.85i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.97 - 1.44i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.312 + 1.77i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.27 + 1.55i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.56 - 3.83i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.96 - 8.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (7.06 + 12.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.59 - 1.34i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.27 - 18.6i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.65105386894921549614427941288, −9.762074525931076964765548749341, −8.624929701259519216600021184858, −8.138160875861932071273381547460, −6.48749049166393789296893687971, −5.86100626602570387874992370843, −4.87299156287759631376312015327, −3.56532544479615363107404416225, −2.27455410112133623969844119438, −0.881823233879708215248869544484,
1.65783100539560056347281473378, 3.76892810862553527844065442750, 4.42839003622035276508453495277, 5.69053152273040528359961803401, 6.38754244502390357045542452979, 7.26542754740591909612757986658, 8.335294967279998006820767553508, 9.318575752747617627664596049229, 10.03898803713775102193425892679, 10.95345524740534088328464423514
