L(s) = 1 | + 2-s + (1.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (1.5 + 0.866i)6-s + (−2.5 + 0.866i)7-s + 8-s + (1.5 + 2.59i)9-s + (−0.5 + 0.866i)10-s + (1 + 1.73i)11-s + (1.5 + 0.866i)12-s + (1 + 1.73i)13-s + (−2.5 + 0.866i)14-s + (−1.5 + 0.866i)15-s + 16-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.866 + 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (−0.158 + 0.273i)10-s + (0.301 + 0.522i)11-s + (0.433 + 0.249i)12-s + (0.277 + 0.480i)13-s + (−0.668 + 0.231i)14-s + (−0.387 + 0.223i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.27381 + 1.46187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27381 + 1.46187i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 13T + 47T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + (-2.5 + 4.33i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.62962153097250828872004029468, −9.929644513263624698738613915595, −9.083583694214073728817456659994, −8.168145284298224933499347739887, −7.01592005230720970635549486663, −6.38636564835284106031753590309, −5.00832588419361105207795447674, −4.02214359256244663574403149857, −3.18908941618452744115193526075, −2.18796842113466365255966777146,
1.19621309978883793966008956975, 2.94440661227173380044468125162, 3.53188262464254293350454723829, 4.70482310153202224464938508039, 6.10417386789229516991477720880, 6.75177669608496408695094495100, 7.78694399837654412450307469086, 8.561971680372789045635685067720, 9.538689465125419467724415007987, 10.36160466717368950842345092319