| L(s) = 1 | + 2-s + 4-s − i·5-s + 4.60i·7-s + 8-s − i·10-s − 0.605·13-s + 4.60i·14-s + 16-s + (3.60 + 2i)17-s − i·20-s − 2i·23-s − 25-s − 0.605·26-s + 4.60i·28-s + 7.21i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447i·5-s + 1.74i·7-s + 0.353·8-s − 0.316i·10-s − 0.167·13-s + 1.23i·14-s + 0.250·16-s + (0.874 + 0.485i)17-s − 0.223i·20-s − 0.417i·23-s − 0.200·25-s − 0.118·26-s + 0.870i·28-s + 1.33i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550732121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550732121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 + (-3.60 - 2i)T \) |
| good | 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 1.39iT - 41T^{2} \) |
| 43 | \( 1 + 2.60T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 3.21iT - 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 6.60iT - 73T^{2} \) |
| 79 | \( 1 + 7.21iT - 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 0.788T + 89T^{2} \) |
| 97 | \( 1 + 2.60iT - 97T^{2} \) |
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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
−9.580540816955116343665947450593, −8.610736297240348097137454642347, −8.234083591654597697027333327561, −6.98079468157561921750477977511, −6.12112443919118678170337812687, −5.35943655463496157055845675431, −4.85535854793417922484943498820, −3.53071695182181102315133001049, −2.65897925674873127236371575395, −1.57423381980619457902063270129,
0.820428302513745903856681910079, 2.32543288150340438539571222722, 3.57946978313510230279863335062, 4.04647076118375761473190963547, 5.10708418917048341508028319892, 6.03861950899744240786195166765, 7.05118551959432972900704322490, 7.40684224643034359111711194742, 8.237979723403437760466032153272, 9.773992816403931560048296534242
