L(s) = 1 | + 2·4-s − 2i·7-s − 3·11-s − 4i·13-s + 4·16-s + 6i·17-s + 19-s − 6i·23-s − 4i·28-s + 9·29-s − 31-s − 8i·37-s + 3·41-s − 4i·43-s − 6·44-s + ⋯ |
L(s) = 1 | + 4-s − 0.755i·7-s − 0.904·11-s − 1.10i·13-s + 16-s + 1.45i·17-s + 0.229·19-s − 1.25i·23-s − 0.755i·28-s + 1.67·29-s − 0.179·31-s − 1.31i·37-s + 0.468·41-s − 0.609i·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.073835067\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.073835067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.848148631199312682378781170908, −7.978920138078111384126912114818, −7.58984611297168225253388670802, −6.61909646144205238113422917343, −5.96480758774958513175120158572, −5.07420871472195861382190796192, −3.94095302669928419138325805927, −3.01658162898504592438145402597, −2.11370439186242015492247528047, −0.73804223298229897906345000775,
1.38758845020114474249724889051, 2.58147644182634272158888303380, 3.04401854990106258467557738750, 4.54310381076291900501773236270, 5.35579023265781539742934366489, 6.17209209093396426575799248006, 6.96213152393121783490717871446, 7.62706101882939484513659761584, 8.421563154696648728197810734967, 9.409265076593214826447202028670