L(s) = 1 | + (−0.866 − 0.5i)3-s + (2.23 − 0.133i)5-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s + (−1.99 − i)15-s + (3.46 + 2i)17-s + (−3.46 + 2i)23-s + (4.96 − 0.598i)25-s − 0.999i·27-s + 6·29-s + (2 − 3.46i)31-s + (−3.46 + 1.99i)33-s + (−6.92 + 4i)37-s + 10·41-s − 4i·43-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (0.998 − 0.0599i)5-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.516 − 0.258i)15-s + (0.840 + 0.485i)17-s + (−0.722 + 0.417i)23-s + (0.992 − 0.119i)25-s − 0.192i·27-s + 1.11·29-s + (0.359 − 0.622i)31-s + (−0.603 + 0.348i)33-s + (−1.13 + 0.657i)37-s + 1.56·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.034904090\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.034904090\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (-3.46 + 2i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-6.92 - 4i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.624807293544545280573865391233, −8.000312433135277015533915999809, −6.97676365968427584108296907095, −6.17691336383326233273012993272, −5.82680832856455632703707059416, −4.99568202989921654313559139755, −3.90947180048060406283874341320, −2.91588936188680976752005186297, −1.77386860575861602152715686459, −0.845782961258970019901223728095,
1.07331242778245243522132327064, 2.11794387766076669046191034499, 3.16339255585273611147973998494, 4.34758824835669358294309739775, 4.95794196273389300800434867635, 5.82845157108284219333384889279, 6.46982464763614533738422632785, 7.15769168720459446717107125232, 8.071694845612806022153552016698, 9.148473332441876864083001953283