L(s) = 1 | + (1 − 1.41i)3-s + 1.41i·5-s + 4·7-s + (−1.00 − 2.82i)9-s − 5.65i·11-s + 4.24i·13-s + (2.00 + 1.41i)15-s − 2.82i·17-s + (−1 + 4.24i)19-s + (4 − 5.65i)21-s − 1.41i·23-s + 2.99·25-s + (−5.00 − 1.41i)27-s + 6·29-s − 4.24i·31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + 0.632i·5-s + 1.51·7-s + (−0.333 − 0.942i)9-s − 1.70i·11-s + 1.17i·13-s + (0.516 + 0.365i)15-s − 0.685i·17-s + (−0.229 + 0.973i)19-s + (0.872 − 1.23i)21-s − 0.294i·23-s + 0.599·25-s + (−0.962 − 0.272i)27-s + 1.11·29-s − 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02621 - 0.913319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02621 - 0.913319i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
| 19 | \( 1 + (1 - 4.24i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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
−9.978671065961800473017045413671, −8.668407020429631791837187438692, −8.444961022619276800782710193718, −7.51681246314878098211096912736, −6.64270932641587577872259995017, −5.84669845906003147891072410212, −4.57640624267292969372136466235, −3.38603598172315336102113580230, −2.34271117101274267172122724383, −1.17183039492744993515610633033,
1.55648271257340826115003204993, 2.70008758134688100532996270514, 4.19956131266350948543110261800, 4.82959454165294789558560807376, 5.36109871006620078070576367038, 7.05096874502569349896050384171, 7.975050154688608856900829809602, 8.485436469931972758114408459179, 9.300069343039410276111258811739, 10.28509335105820461948263180658