| L(s) = 1 | + (0.463 − 2.92i)2-s + (−0.866 + 0.441i)3-s + (−4.52 − 1.47i)4-s + (0.953 + 4.90i)5-s + (0.889 + 2.73i)6-s + (4.44 − 4.44i)7-s + (−1.02 + 2.00i)8-s + (−4.73 + 6.51i)9-s + (14.7 − 0.516i)10-s + (−9.24 + 6.71i)11-s + (4.57 − 0.724i)12-s + (0.202 + 1.27i)13-s + (−10.9 − 15.0i)14-s + (−2.99 − 3.83i)15-s + (−10.0 − 7.27i)16-s + (−22.3 − 11.3i)17-s + ⋯ |
| L(s) = 1 | + (0.231 − 1.46i)2-s + (−0.288 + 0.147i)3-s + (−1.13 − 0.367i)4-s + (0.190 + 0.981i)5-s + (0.148 + 0.456i)6-s + (0.635 − 0.635i)7-s + (−0.127 + 0.250i)8-s + (−0.525 + 0.723i)9-s + (1.47 − 0.0516i)10-s + (−0.840 + 0.610i)11-s + (0.381 − 0.0603i)12-s + (0.0155 + 0.0981i)13-s + (−0.781 − 1.07i)14-s + (−0.199 − 0.255i)15-s + (−0.626 − 0.454i)16-s + (−1.31 − 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.743371 - 0.610646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.743371 - 0.610646i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.953 - 4.90i)T \) |
| good | 2 | \( 1 + (-0.463 + 2.92i)T + (-3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.441i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-4.44 + 4.44i)T - 49iT^{2} \) |
| 11 | \( 1 + (9.24 - 6.71i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-0.202 - 1.27i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (22.3 + 11.3i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-31.7 + 10.3i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-9.63 - 1.52i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (-13.6 - 4.42i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (3.95 + 12.1i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-32.7 + 5.18i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-34.5 - 25.0i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (10.6 + 10.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (17.0 + 33.4i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (66.4 - 33.8i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-21.1 + 29.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (9.86 - 7.17i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-42.3 - 21.5i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (18.5 - 57.2i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-40.6 - 6.43i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (66.7 + 21.6i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (14.9 - 29.3i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-23.3 - 32.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (16.1 + 31.6i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−17.64013410282397264509923713579, −15.82666935068457551314113624013, −14.11947161670006241707247609418, −13.28564077089421007553804900188, −11.42011299492660153784386968456, −10.96784572724263643653973576020, −9.777839213448888563924275207212, −7.36746317318344865732743893761, −4.80284693201528656119928315354, −2.62540044968977132192978421769,
5.07602454627440251085490671371, 6.07047243393144782957587279356, 8.007002550218209006739451692937, 8.989223880633309605412700340613, 11.43016997675443249597672195077, 12.91744781621348919389323564295, 14.16715210340377355058184978462, 15.43018197378797499972656926406, 16.27618399600207413754345744646, 17.49005569964205280469619611231
