L(s) = 1 | + (0.5 − 0.866i)2-s + (−1.31 + 1.13i)3-s + (−0.499 − 0.866i)4-s + (1.08 + 0.627i)5-s + (0.326 + 1.70i)6-s + (1.14 + 2.38i)7-s − 0.999·8-s + (0.432 − 2.96i)9-s + (1.08 − 0.627i)10-s + (−0.900 − 1.56i)11-s + (1.63 + 0.568i)12-s + (−0.657 + 3.54i)13-s + (2.63 + 0.204i)14-s + (−2.13 + 0.409i)15-s + (−0.5 + 0.866i)16-s + (1.12 + 1.95i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.756 + 0.654i)3-s + (−0.249 − 0.433i)4-s + (0.486 + 0.280i)5-s + (0.133 + 0.694i)6-s + (0.431 + 0.902i)7-s − 0.353·8-s + (0.144 − 0.989i)9-s + (0.343 − 0.198i)10-s + (−0.271 − 0.470i)11-s + (0.472 + 0.164i)12-s + (−0.182 + 0.983i)13-s + (0.704 + 0.0545i)14-s + (−0.551 + 0.105i)15-s + (−0.125 + 0.216i)16-s + (0.273 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23195 + 0.556932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23195 + 0.556932i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (1.31 - 1.13i)T \) |
| 7 | \( 1 + (-1.14 - 2.38i)T \) |
| 13 | \( 1 + (0.657 - 3.54i)T \) |
good | 5 | \( 1 + (-1.08 - 0.627i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.900 + 1.56i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.12 - 1.95i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.46 - 4.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.93 - 4.58i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.62iT - 29T^{2} \) |
| 31 | \( 1 + (-2.39 - 4.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.77 + 2.75i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.52iT - 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 + (-8.20 - 4.73i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.2 + 6.52i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.80 - 2.77i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 1.52i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.4 - 6.60i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 + (7.95 + 13.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.323 - 0.560i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + (-11.1 - 6.42i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.48T + 97T^{2} \) |
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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
−10.69915198566393046406473177631, −10.50083490127244028411219917798, −9.214334153614561924329557075432, −8.756583752530527931126793105639, −7.05004758642213421071939676971, −5.89021707269638724534620375106, −5.40317346189444405507196175869, −4.30533334238521434909720216028, −3.12086064032729403717962892042, −1.68430431240835531386615989220,
0.814030117408671583112464598708, 2.60330907778161076094974393176, 4.48435681020459073595690013340, 5.11291448615058020121038317214, 6.08388367259455386391762638579, 7.11542656336546683854484578313, 7.61493906316129785375276857424, 8.666970509405201587015107258868, 9.951721934140575731466980004768, 10.74722900925990318360759305772