| L(s) = 1 | + (0.219 − 0.379i)2-s + (−3.77 − 3.57i)3-s + (3.90 + 6.76i)4-s − 16.0·5-s + (−2.18 + 0.649i)6-s + (−1.97 + 18.4i)7-s + 6.93·8-s + (1.47 + 26.9i)9-s + (−3.52 + 6.11i)10-s − 26.5·11-s + (9.42 − 39.4i)12-s + (−7.29 + 12.6i)13-s + (6.56 + 4.78i)14-s + (60.7 + 57.4i)15-s + (−29.7 + 51.4i)16-s + (−17.3 + 29.9i)17-s + ⋯ |
| L(s) = 1 | + (0.0775 − 0.134i)2-s + (−0.726 − 0.687i)3-s + (0.487 + 0.845i)4-s − 1.43·5-s + (−0.148 + 0.0442i)6-s + (−0.106 + 0.994i)7-s + 0.306·8-s + (0.0546 + 0.998i)9-s + (−0.111 + 0.193i)10-s − 0.727·11-s + (0.226 − 0.949i)12-s + (−0.155 + 0.269i)13-s + (0.125 + 0.0913i)14-s + (1.04 + 0.989i)15-s + (−0.464 + 0.804i)16-s + (−0.247 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.236000 + 0.438203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.236000 + 0.438203i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.77 + 3.57i)T \) |
| 7 | \( 1 + (1.97 - 18.4i)T \) |
| good | 2 | \( 1 + (-0.219 + 0.379i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 16.0T + 125T^{2} \) |
| 11 | \( 1 + 26.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + (7.29 - 12.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (17.3 - 29.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (65.3 + 113. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 66.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + (102. + 177. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-126. - 219. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-111. - 193. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-42.4 + 73.6i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (16.6 + 28.8i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (262. - 454. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (196. - 339. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-324. - 562. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (19.3 - 33.5i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-90.1 - 156. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 624.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-155. + 270. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-243. + 421. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-345. - 598. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (525. + 909. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (86.6 + 149. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.30811103055011137920981486283, −13.18316027000274264519770234843, −12.43876236706144302399026197857, −11.61007497918909579127841897190, −10.95238793295849100639634316883, −8.551974502457585934377449827035, −7.65638245888619026544286520069, −6.53230284771353996914864796020, −4.63080985482902239038217390735, −2.67105605842762025251347147203,
0.34890398827985906868128442709, 3.82723381658482407020417556552, 5.10478429517924250883918533210, 6.68233677030499504877313925296, 7.86886754808422188926650847484, 9.863520272050639871695613706196, 10.81097003903072779093556802540, 11.44774696632053332466356604227, 12.79514570860137700152315938838, 14.54874125225580969402520942408
