| L(s) = 1 | + (−1.67 + 0.437i)3-s + (−2.23 − 1.41i)7-s + (2.61 − 1.46i)9-s − 3.83i·11-s − 5.99i·13-s − 2.07·17-s − 5.11i·19-s + (4.36 + 1.39i)21-s + 8.57i·23-s + (−3.74 + 3.59i)27-s + 5.64i·29-s + 1.95i·31-s + (1.67 + 6.42i)33-s − 2.23·37-s + (2.61 + 10.0i)39-s + ⋯ |
| L(s) = 1 | + (−0.967 + 0.252i)3-s + (−0.845 − 0.534i)7-s + (0.872 − 0.488i)9-s − 1.15i·11-s − 1.66i·13-s − 0.502·17-s − 1.17i·19-s + (0.952 + 0.303i)21-s + 1.78i·23-s + (−0.721 + 0.692i)27-s + 1.04i·29-s + 0.351i·31-s + (0.291 + 1.11i)33-s − 0.367·37-s + (0.419 + 1.60i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1977352697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1977352697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 0.437i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 11 | \( 1 + 3.83iT - 11T^{2} \) |
| 13 | \( 1 + 5.99iT - 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 5.11iT - 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 - 5.64iT - 29T^{2} \) |
| 31 | \( 1 - 1.95iT - 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 4.37iT - 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 2.02iT - 71T^{2} \) |
| 73 | \( 1 - 1.41iT - 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 + 0.206iT - 97T^{2} \) |
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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
−8.835650884244751597615468646262, −7.72070719858383740165491862479, −7.04352412813852458888609190584, −6.17197039418732489038894505110, −5.57589058805129244470365644671, −4.79246384074168861547138592228, −3.57817114874389099903334077821, −3.05074004787346537481831817086, −1.09278817934115324848362527207, −0.090769190744340710594804952334,
1.68511773580890735873238225823, 2.53952806765960492819335177747, 4.27903193236050252127232882098, 4.46143491098909595235074804210, 5.85345943697418855385841260533, 6.34192559751999276399981439058, 6.97896748078818253651042672148, 7.79730732520203250421768592586, 8.938511328345185467340362170095, 9.595867579061989785519728107490
