| L(s) = 1 | − i·3-s − i·7-s − 9-s + 4·11-s + 3i·13-s + 4i·17-s − 19-s − 21-s + i·27-s − 8·29-s + 31-s − 4i·33-s + 2i·37-s + 3·39-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.20·11-s + 0.832i·13-s + 0.970i·17-s − 0.229·19-s − 0.218·21-s + 0.192i·27-s − 1.48·29-s + 0.179·31-s − 0.696i·33-s + 0.328i·37-s + 0.480·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435124514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435124514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 11iT - 97T^{2} \) |
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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
−8.337108895267978995840032994118, −7.62188589956049523086289281434, −6.95227893350302357251715214884, −6.27577456833496417497671371388, −5.77674382799543019999448447047, −4.46736326929441130098401667907, −4.03011502630606505203535847320, −3.03376651078939901368726059557, −1.83899806843930016478444783529, −1.21479060369737038656179596523,
0.40199642550902986005306749672, 1.78634463117542396270569526943, 2.83744380917788551993849211975, 3.66570955847874630455957625159, 4.33929041074983240665998587469, 5.34284728186920593920297000209, 5.76150798935628351367686174381, 6.74837415113636248996016409445, 7.38693308019363150879508646591, 8.290585562573321736942416223792
