L(s) = 1 | + (0.328 − 1.37i)2-s + (−1.78 − 0.903i)4-s − 0.512i·5-s − 7-s + (−1.82 + 2.15i)8-s + (−0.704 − 0.168i)10-s + 1.82i·11-s + 1.80i·13-s + (−0.328 + 1.37i)14-s + (2.36 + 3.22i)16-s + 8.11·17-s + 3.43i·19-s + (−0.462 + 0.914i)20-s + (2.51 + 0.600i)22-s − 3.65·23-s + ⋯ |
L(s) = 1 | + (0.232 − 0.972i)2-s + (−0.892 − 0.451i)4-s − 0.229i·5-s − 0.377·7-s + (−0.646 + 0.763i)8-s + (−0.222 − 0.0531i)10-s + 0.551i·11-s + 0.500i·13-s + (−0.0877 + 0.367i)14-s + (0.592 + 0.805i)16-s + 1.96·17-s + 0.789i·19-s + (−0.103 + 0.204i)20-s + (0.536 + 0.128i)22-s − 0.762·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585128524\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585128524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.328 + 1.37i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 0.512iT - 5T^{2} \) |
| 11 | \( 1 - 1.82iT - 11T^{2} \) |
| 13 | \( 1 - 1.80iT - 13T^{2} \) |
| 17 | \( 1 - 8.11T + 17T^{2} \) |
| 19 | \( 1 - 3.43iT - 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 7.98iT - 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.87iT - 53T^{2} \) |
| 59 | \( 1 - 2.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.31iT - 61T^{2} \) |
| 67 | \( 1 - 6.44iT - 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 - 4.53iT - 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.470883127960428465395749789233, −8.899689024552940849633244095111, −7.904773579136874401540206681297, −6.99897971821877338548665524584, −5.74671120993647358573458312141, −5.22549980261484405888328461660, −4.03570541502080047752551167212, −3.40698414807423521583539643021, −2.18876704322959046306628229030, −1.07183800269576097156063959561,
0.74454432723300926003519802207, 2.88509891759936813643539302712, 3.58505425836269565315796404151, 4.72436959861321330444494608672, 5.63700673628159355195280635446, 6.23177279003531010599862236563, 7.12235106359541059531685922917, 7.958166838273666566753683465398, 8.448028844569745291509171229299, 9.626277398638281055395839767942