| L(s) = 1 | + 2-s + (1 − 1.41i)3-s + 4-s + i·5-s + (1 − 1.41i)6-s + 0.585·7-s + 8-s + (−1.00 − 2.82i)9-s + i·10-s − 5.41i·11-s + (1 − 1.41i)12-s + 2.24i·13-s + 0.585·14-s + (1.41 + i)15-s + 16-s − 2.82i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.577 − 0.816i)3-s + 0.5·4-s + 0.447i·5-s + (0.408 − 0.577i)6-s + 0.221·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s + 0.316i·10-s − 1.63i·11-s + (0.288 − 0.408i)12-s + 0.621i·13-s + 0.156·14-s + (0.365 + 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46086 - 1.10923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46086 - 1.10923i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-4.24 - i)T \) |
| good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 5.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 - 2.24iT - 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 3.17iT - 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 6.58iT - 97T^{2} \) |
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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
−10.96389387510736316835769717218, −9.676950236032948983858772177977, −8.683228756300927528409459671252, −7.82139138698658946244817395544, −6.93632721454390896411354400547, −6.15120139300849228076225475644, −5.13496602190129717167897367182, −3.51427677932967976688097641075, −2.94739214611423107813885716974, −1.39619237750898003875282140663,
1.98662185733131327668353988066, 3.19322445155388735948479048089, 4.48588244476838388878468021497, 4.83505278822222212966969056197, 6.07107195804993533841591465537, 7.41776250938688509094261352850, 8.138600753278654802567272390653, 9.221640075872049327016954973952, 10.05826051264888194840558233165, 10.70265913806466196441551879159
