| L(s) = 1 | + (−1.40 − 0.197i)2-s + (−1.23 + 2.98i)3-s + (1.92 + 0.552i)4-s + (0.453 − 0.891i)5-s + (2.32 − 3.93i)6-s + (0.479 + 1.99i)7-s + (−2.58 − 1.15i)8-s + (−5.26 − 5.26i)9-s + (−0.811 + 1.15i)10-s + (4.08 − 4.78i)11-s + (−4.02 + 5.05i)12-s + (1.84 + 3.00i)13-s + (−0.277 − 2.89i)14-s + (2.09 + 2.45i)15-s + (3.38 + 2.12i)16-s + (0.0405 + 0.515i)17-s + ⋯ |
| L(s) = 1 | + (−0.990 − 0.139i)2-s + (−0.714 + 1.72i)3-s + (0.961 + 0.276i)4-s + (0.203 − 0.398i)5-s + (0.947 − 1.60i)6-s + (0.181 + 0.755i)7-s + (−0.913 − 0.407i)8-s + (−1.75 − 1.75i)9-s + (−0.256 + 0.366i)10-s + (1.23 − 1.44i)11-s + (−1.16 + 1.45i)12-s + (0.510 + 0.833i)13-s + (−0.0741 − 0.773i)14-s + (0.542 + 0.634i)15-s + (0.847 + 0.531i)16-s + (0.00983 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528428 + 0.693885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528428 + 0.693885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.197i)T \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
| 41 | \( 1 + (-3.56 + 5.31i)T \) |
| good | 3 | \( 1 + (1.23 - 2.98i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.479 - 1.99i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-4.08 + 4.78i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.84 - 3.00i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.0405 - 0.515i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-6.16 - 3.77i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-4.23 - 3.07i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.238 - 3.02i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.35 - 7.23i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.10 + 6.46i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (1.04 + 6.56i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.936 + 3.90i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.00635 + 0.000499i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (8.66 - 11.9i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.886 - 5.59i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (4.19 - 3.57i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-4.57 - 3.91i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (3.95 - 3.95i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.675 - 0.279i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 5.73iT - 83T^{2} \) |
| 89 | \( 1 + (1.99 - 0.479i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-4.55 + 3.88i)T + (15.1 - 95.8i)T^{2} \) |
| 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
−10.50381728652972073530227265394, −9.439816317438204246546911849428, −8.924997087032960826834484144690, −8.697906386230053875262216149681, −6.96858997465874995387337886241, −5.73817051704013303895587970565, −5.54635623357776838513055443018, −3.90933410294295580816728330204, −3.26050198067008572089805811979, −1.21801472685605767401280692168,
0.811515355428333898302628587275, 1.64869760739219878144619799828, 2.89575777018174392141899923893, 4.93394917274149383420344965200, 6.22193082168844372032679745214, 6.64416471276350563759796053223, 7.55404254371511044948971113445, 7.73560149396393153828936313109, 9.132735463231110531708999849903, 9.964141582495993582151712377045
