| L(s) = 1 | + (0.909 + 1.08i)2-s + (−0.830 + 2.28i)3-s + (−0.347 + 1.96i)4-s + (0.0233 + 0.132i)5-s + (−3.22 + 1.17i)6-s + (4.79 − 8.30i)7-s + (−2.44 + 1.41i)8-s + (2.37 + 1.99i)9-s + (−0.122 + 0.145i)10-s + (−7.19 − 12.4i)11-s + (−4.20 − 2.42i)12-s + (1.30 + 3.57i)13-s + (13.3 − 2.35i)14-s + (−0.321 − 0.0567i)15-s + (−3.75 − 1.36i)16-s + (−3.78 + 3.17i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.276 + 0.760i)3-s + (−0.0868 + 0.492i)4-s + (0.00467 + 0.0264i)5-s + (−0.538 + 0.195i)6-s + (0.684 − 1.18i)7-s + (−0.306 + 0.176i)8-s + (0.263 + 0.221i)9-s + (−0.0122 + 0.0145i)10-s + (−0.653 − 1.13i)11-s + (−0.350 − 0.202i)12-s + (0.100 + 0.274i)13-s + (0.953 − 0.168i)14-s + (−0.0214 − 0.00378i)15-s + (−0.234 − 0.0855i)16-s + (−0.222 + 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02245 + 0.662483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02245 + 0.662483i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 19 | \( 1 + (17.3 + 7.75i)T \) |
| good | 3 | \( 1 + (0.830 - 2.28i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (-0.0233 - 0.132i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.79 + 8.30i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.19 + 12.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.30 - 3.57i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (3.78 - 3.17i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (3.30 - 18.7i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (23.8 - 28.3i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-26.2 - 15.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 48.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (9.73 - 26.7i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (10.5 + 59.8i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (48.6 + 40.8i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-31.2 - 5.51i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-54.6 - 65.1i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (20.0 - 113. i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-42.6 + 50.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-56.2 + 9.91i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-75.4 - 27.4i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (35.2 - 96.9i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (40.5 - 70.2i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (42.3 + 116. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-6.38 - 7.61i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−16.35129748107499046114788576866, −15.22975553032035584302489656738, −14.00484243899087212157173783178, −13.11097587621821804385778129494, −11.18547671794325085087985589465, −10.44328603920539560446949397105, −8.529979915039901214481483935908, −7.10972697398981605946671533082, −5.25619649192932036697397235611, −3.97988591769579696185072244645,
2.14055643804036482267076325834, 4.82817256802283830953721226733, 6.39447917743742527164111074914, 8.133477904362590360961291745885, 9.826517233700153172143516542982, 11.37743424123981741539925631352, 12.42328916114169550908561254484, 13.04146857415836855671917837326, 14.76004258167089230334739826349, 15.45679508636638482566236660744
