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
−8.368377662090751794568123209906, −7.48775863328630420677263105901, −6.95518668302474737944403125432, −6.07470980288019299768386950269, −5.53368396356124246064389418368, −4.66488216391526833116132786074, −4.04549234059031689758652238633, −3.00002392918455765978106957704, −1.71058066747505402501371094740, −0.64929992173998420312843924500,
0.70403128170767144417926310928, 2.09306988315601827752931740961, 2.65144502771125855883790809772, 4.10668322309465886063305640343, 4.92729515322644886300228581782, 5.57829561969687322177222129897, 6.15632300303431491451162882332, 6.81002559804256443303794448643, 7.75128842437737612549537815128, 8.347253574094802884659720569667