| L(s) = 1 | + (−1.68 − 0.417i)3-s + 5-s + 2.42i·7-s + (2.65 + 1.40i)9-s − 2.52·11-s + 0.420i·13-s + (−1.68 − 0.417i)15-s − 1.53i·17-s − 1.99·19-s + (1.01 − 4.08i)21-s − 2.90i·23-s + 25-s + (−3.87 − 3.46i)27-s − 1.32i·29-s − 4.77i·31-s + ⋯ |
| L(s) = 1 | + (−0.970 − 0.240i)3-s + 0.447·5-s + 0.917i·7-s + (0.883 + 0.467i)9-s − 0.762·11-s + 0.116i·13-s + (−0.434 − 0.107i)15-s − 0.371i·17-s − 0.456·19-s + (0.221 − 0.890i)21-s − 0.606i·23-s + 0.200·25-s + (−0.745 − 0.666i)27-s − 0.245i·29-s − 0.856i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259889378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259889378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 + 0.417i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-7.34 + 3.61i)T \) |
| good | 7 | \( 1 - 2.42iT - 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 - 0.420iT - 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 2.90iT - 23T^{2} \) |
| 29 | \( 1 + 1.32iT - 29T^{2} \) |
| 31 | \( 1 + 4.77iT - 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 - 2.00T + 41T^{2} \) |
| 43 | \( 1 - 2.63iT - 43T^{2} \) |
| 47 | \( 1 + 2.60iT - 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + 14.2iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 71 | \( 1 + 3.60iT - 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 7.44iT - 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.46iT - 89T^{2} \) |
| 97 | \( 1 - 4.51iT - 97T^{2} \) |
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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
−8.347253574094802884659720569667, −7.75128842437737612549537815128, −6.81002559804256443303794448643, −6.15632300303431491451162882332, −5.57829561969687322177222129897, −4.92729515322644886300228581782, −4.10668322309465886063305640343, −2.65144502771125855883790809772, −2.09306988315601827752931740961, −0.70403128170767144417926310928,
0.64929992173998420312843924500, 1.71058066747505402501371094740, 3.00002392918455765978106957704, 4.04549234059031689758652238633, 4.66488216391526833116132786074, 5.53368396356124246064389418368, 6.07470980288019299768386950269, 6.95518668302474737944403125432, 7.48775863328630420677263105901, 8.368377662090751794568123209906
