| L(s) = 1 | + 1.43i·5-s − 1.94i·7-s − 2.87i·11-s + (1.38 + 3.32i)13-s − 0.495·17-s + 2.71i·19-s − 3.21·23-s + 2.94·25-s + 5.25·29-s + 3.28i·31-s + 2.78·35-s + 2.87i·37-s + 1.77i·41-s − 3.23·43-s − 9.82i·47-s + ⋯ |
| L(s) = 1 | + 0.641i·5-s − 0.735i·7-s − 0.865i·11-s + (0.385 + 0.922i)13-s − 0.120·17-s + 0.623i·19-s − 0.671·23-s + 0.588·25-s + 0.975·29-s + 0.590i·31-s + 0.471·35-s + 0.472i·37-s + 0.277i·41-s − 0.493·43-s − 1.43i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.802614667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802614667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-1.38 - 3.32i)T \) |
| good | 5 | \( 1 - 1.43iT - 5T^{2} \) |
| 7 | \( 1 + 1.94iT - 7T^{2} \) |
| 11 | \( 1 + 2.87iT - 11T^{2} \) |
| 17 | \( 1 + 0.495T + 17T^{2} \) |
| 19 | \( 1 - 2.71iT - 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 5.25T + 29T^{2} \) |
| 31 | \( 1 - 3.28iT - 31T^{2} \) |
| 37 | \( 1 - 2.87iT - 37T^{2} \) |
| 41 | \( 1 - 1.77iT - 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 9.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.86T + 53T^{2} \) |
| 59 | \( 1 + 3.34iT - 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 1.04iT - 67T^{2} \) |
| 71 | \( 1 + 0.437iT - 71T^{2} \) |
| 73 | \( 1 - 3.34iT - 73T^{2} \) |
| 79 | \( 1 - 9.55T + 79T^{2} \) |
| 83 | \( 1 - 5.59iT - 83T^{2} \) |
| 89 | \( 1 - 5.99iT - 89T^{2} \) |
| 97 | \( 1 + 7.25iT - 97T^{2} \) |
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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
−8.616570916569766942157252236551, −8.264432923427487158010799436975, −7.12125713492745307906364159805, −6.68980806563040879301570210244, −5.93450000169552570226866249140, −4.90684203362899164607814234969, −3.92233317821252706341716633830, −3.33599967342735454031640187351, −2.18294886376522358624630763209, −0.931851239984818060651241907996,
0.76541538732837265117026003897, 2.07023961702081859240239195353, 2.93741223629944660339613798612, 4.12175649885999888350885969014, 4.90564742329622559141994931625, 5.59018912645047696106933035360, 6.39521113744955407409871448524, 7.30519034975362604339011308783, 8.132450275251051499573198457237, 8.729380797807198837828352169737
