| L(s) = 1 | + (0.670 + 0.0729i)2-s + (0.865 − 1.01i)3-s + (−1.50 − 0.332i)4-s + (0.331 + 0.251i)5-s + (0.654 − 0.620i)6-s + (−0.928 + 2.32i)7-s + (−2.26 − 0.763i)8-s + (0.196 + 1.19i)9-s + (0.203 + 0.192i)10-s + (−0.551 + 0.255i)11-s + (−1.64 + 1.24i)12-s + (0.568 − 3.46i)13-s + (−0.792 + 1.49i)14-s + (0.542 − 0.119i)15-s + (1.34 + 0.620i)16-s + (−0.0439 − 0.110i)17-s + ⋯ |
| L(s) = 1 | + (0.474 + 0.0515i)2-s + (0.499 − 0.588i)3-s + (−0.754 − 0.166i)4-s + (0.148 + 0.112i)5-s + (0.267 − 0.253i)6-s + (−0.350 + 0.880i)7-s + (−0.801 − 0.269i)8-s + (0.0654 + 0.399i)9-s + (0.0644 + 0.0610i)10-s + (−0.166 + 0.0768i)11-s + (−0.474 + 0.360i)12-s + (0.157 − 0.961i)13-s + (−0.211 + 0.399i)14-s + (0.140 − 0.0308i)15-s + (0.335 + 0.155i)16-s + (−0.0106 − 0.0267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02462 - 0.136425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02462 - 0.136425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (7.64 + 0.742i)T \) |
| good | 2 | \( 1 + (-0.670 - 0.0729i)T + (1.95 + 0.429i)T^{2} \) |
| 3 | \( 1 + (-0.865 + 1.01i)T + (-0.485 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.331 - 0.251i)T + (1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (0.928 - 2.32i)T + (-5.08 - 4.81i)T^{2} \) |
| 11 | \( 1 + (0.551 - 0.255i)T + (7.12 - 8.38i)T^{2} \) |
| 13 | \( 1 + (-0.568 + 3.46i)T + (-12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (0.0439 + 0.110i)T + (-12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (3.52 + 5.20i)T + (-7.03 + 17.6i)T^{2} \) |
| 23 | \( 1 + (0.233 - 4.30i)T + (-22.8 - 2.48i)T^{2} \) |
| 29 | \( 1 + (-7.91 + 0.860i)T + (28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (1.03 - 1.52i)T + (-11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (-2.60 + 0.879i)T + (29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (-0.288 - 5.32i)T + (-40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (9.32 + 4.31i)T + (27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (-1.73 + 1.31i)T + (12.5 - 45.2i)T^{2} \) |
| 53 | \( 1 + (8.32 - 7.88i)T + (2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (-7.26 - 0.790i)T + (59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (8.23 + 2.77i)T + (53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (7.83 - 5.95i)T + (18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (-3.97 + 7.49i)T + (-40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (-5.82 - 6.86i)T + (-12.7 + 77.9i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 7.84i)T + (38.8 - 73.3i)T^{2} \) |
| 89 | \( 1 + (-7.04 + 0.766i)T + (86.9 - 19.1i)T^{2} \) |
| 97 | \( 1 + (0.594 + 1.12i)T + (-54.4 + 80.2i)T^{2} \) |
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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
−14.96418721853170166446058846187, −13.75563948064412049072140948326, −13.12052397132779010790420330380, −12.19405973471738995624485125930, −10.40984364526798053049681897754, −9.061988029191077033555773259995, −8.060379337410594781089362374858, −6.29149217070053306538497656217, −4.92282695630098200676788024041, −2.82688970011499024535134768323,
3.55627553287675980416698105984, 4.52891772087938939049707106560, 6.42343426849875804396841920078, 8.319313726614730074663980138251, 9.390931126613552881967445132765, 10.36169063063355025565851776038, 12.09038379199153120836532797174, 13.17990067662248949071152391280, 14.13491818655522501276092319212, 14.84643746927282491220650314760
