L(s) = 1 | + (1.38 + 0.273i)2-s + (0.758 + 1.55i)3-s + (1.85 + 0.758i)4-s + (0.626 + 2.36i)6-s − 3.56·7-s + (2.36 + 1.55i)8-s + (−1.85 + 2.36i)9-s + 4.20·11-s + (0.222 + 3.45i)12-s − 2.70i·13-s + (−4.94 − 0.973i)14-s + (2.85 + 2.80i)16-s − 0.828·17-s + (−3.21 + 2.77i)18-s − 5.07i·19-s + ⋯ |
L(s) = 1 | + (0.981 + 0.193i)2-s + (0.437 + 0.899i)3-s + (0.925 + 0.379i)4-s + (0.255 + 0.966i)6-s − 1.34·7-s + (0.834 + 0.550i)8-s + (−0.616 + 0.787i)9-s + 1.26·11-s + (0.0642 + 0.997i)12-s − 0.749i·13-s + (−1.32 − 0.260i)14-s + (0.712 + 0.701i)16-s − 0.200·17-s + (−0.757 + 0.653i)18-s − 1.16i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00600 + 1.33067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00600 + 1.33067i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.273i)T \) |
| 3 | \( 1 + (-0.758 - 1.55i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 2.70iT - 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 5.07iT - 19T^{2} \) |
| 23 | \( 1 - 1.09iT - 23T^{2} \) |
| 29 | \( 1 + 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 6.59iT - 31T^{2} \) |
| 37 | \( 1 - 5.40iT - 37T^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 - 0.531T + 43T^{2} \) |
| 47 | \( 1 + 6.22iT - 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.701T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 7.70iT - 73T^{2} \) |
| 79 | \( 1 - 7.12iT - 79T^{2} \) |
| 83 | \( 1 - 3.11iT - 83T^{2} \) |
| 89 | \( 1 - 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 8.10iT - 97T^{2} \) |
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
−11.91885763963778460268786481638, −11.13780426168618694097339142862, −9.979971300349322671176263854534, −9.277857499937942905065502781342, −8.043489710101493088101844668593, −6.74176014595433521714950298798, −5.92258015159152407514844995384, −4.60352623886763456176305167033, −3.62812452581344428518156922242, −2.72226255833081981477679406312,
1.64303959709078520672530672139, 3.14565147961221435013572818879, 4.00548867640619918492761242357, 5.83393443555252540162556797936, 6.62466383474547701825425614472, 7.20797580207077564331214523432, 8.794036224149181509700852643137, 9.655546046383202323424093788153, 10.89018542740280362001261246653, 12.17695123055090941319069486675