| 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
−15.45679508636638482566236660744, −14.76004258167089230334739826349, −13.04146857415836855671917837326, −12.42328916114169550908561254484, −11.37743424123981741539925631352, −9.826517233700153172143516542982, −8.133477904362590360961291745885, −6.39447917743742527164111074914, −4.82817256802283830953721226733, −2.14055643804036482267076325834,
3.97988591769579696185072244645, 5.25619649192932036697397235611, 7.10972697398981605946671533082, 8.529979915039901214481483935908, 10.44328603920539560446949397105, 11.18547671794325085087985589465, 13.11097587621821804385778129494, 14.00484243899087212157173783178, 15.22975553032035584302489656738, 16.35129748107499046114788576866
