L(s) = 1 | + (−1.66 − 1.66i)3-s − 2.89i·7-s + 2.53i·9-s + (−1.84 + 1.84i)11-s + (3.08 + 3.08i)13-s − 7.29·17-s + (1.23 + 1.23i)19-s + (−4.81 + 4.81i)21-s + 4.60i·23-s + (−0.772 + 0.772i)27-s + (4.24 + 4.24i)29-s − 2.06·31-s + 6.13·33-s + (1.17 − 1.17i)37-s − 10.2i·39-s + ⋯ |
L(s) = 1 | + (−0.960 − 0.960i)3-s − 1.09i·7-s + 0.845i·9-s + (−0.556 + 0.556i)11-s + (0.854 + 0.854i)13-s − 1.77·17-s + (0.283 + 0.283i)19-s + (−1.05 + 1.05i)21-s + 0.960i·23-s + (−0.148 + 0.148i)27-s + (0.788 + 0.788i)29-s − 0.370·31-s + 1.06·33-s + (0.193 − 0.193i)37-s − 1.64i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6357243822\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6357243822\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.66 + 1.66i)T + 3iT^{2} \) |
| 7 | \( 1 + 2.89iT - 7T^{2} \) |
| 11 | \( 1 + (1.84 - 1.84i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.08 - 3.08i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + (-1.23 - 1.23i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.60iT - 23T^{2} \) |
| 29 | \( 1 + (-4.24 - 4.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + (-1.17 + 1.17i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.61iT - 41T^{2} \) |
| 43 | \( 1 + (-3.03 + 3.03i)T - 43iT^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + (2.73 - 2.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.11 - 3.11i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.34 - 2.34i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.24 - 8.24i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.25iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 0.113T + 79T^{2} \) |
| 83 | \( 1 + (-9.76 - 9.76i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.74iT - 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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
−9.534336153292828810711881408255, −8.631572804447967944881588435170, −7.60446944636975197974347767384, −6.93869061965885125324023057534, −6.54340426020847544458262749717, −5.52750769361810520513775486552, −4.57316748898101601850384197797, −3.69472096177913353912555727782, −2.07104443206884888562458387018, −1.09948250765368499163253185177,
0.31729965873638083426994024406, 2.32013611511606535058047697929, 3.34625037381025477035901875532, 4.60765375523535899251874365777, 5.07548597902655547415805120588, 6.12196065258091623092945026787, 6.35799843170589529261715579258, 7.975509335892993742652212387750, 8.595544569140524988793453129495, 9.368568436299789236230810182518