L(s) = 1 | + (0.5 + 1.53i)2-s + (0.809 − 0.587i)3-s + (−0.5 + 0.363i)4-s + (1.30 + 0.951i)6-s − 0.618·7-s + (1.80 + 1.31i)8-s + (−0.618 + 1.90i)9-s + (−1.61 − 4.97i)11-s + (−0.190 + 0.587i)12-s + (−0.572 + 1.76i)13-s + (−0.309 − 0.951i)14-s + (−1.50 + 4.61i)16-s + (−4.23 − 3.07i)17-s − 3.23·18-s + (−0.690 − 0.502i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 1.08i)2-s + (0.467 − 0.339i)3-s + (−0.250 + 0.181i)4-s + (0.534 + 0.388i)6-s − 0.233·7-s + (0.639 + 0.464i)8-s + (−0.206 + 0.634i)9-s + (−0.487 − 1.50i)11-s + (−0.0551 + 0.169i)12-s + (−0.158 + 0.489i)13-s + (−0.0825 − 0.254i)14-s + (−0.375 + 1.15i)16-s + (−1.02 − 0.746i)17-s − 0.762·18-s + (−0.158 − 0.115i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26339 + 0.694559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26339 + 0.694559i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.572 - 1.76i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.23 + 3.07i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.690 + 0.502i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.16 + 3.57i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.92 + 2.12i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.42 - 1.76i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.0729 + 0.224i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.236 - 0.726i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.363i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.80 - 2.04i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.35 - 10.3i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.69 - 8.28i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.85 - 2.80i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-5.35 + 3.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.78 - 8.55i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.54 + 4.75i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.04 + 3.66i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.76 + 8.50i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.11 - 2.26i)T + (29.9 - 92.2i)T^{2} \) |
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
−13.80832658042842592661127865303, −13.10227442338752365369864054415, −11.44557043877088879128765738433, −10.57440578657735152882540925382, −8.827104062926329491716274929933, −8.064968574265870627198111806531, −6.94645460846324809154954564180, −5.92607240875110110145163035708, −4.66902844881642411729754878148, −2.59825336491970313666444998998,
2.21996412766724760751917484491, 3.55477354531954723538833763218, 4.70163256403741205678875710037, 6.62837148358234187458345868320, 7.963877643924952661272178780234, 9.457492555984399402444689845909, 10.14941145070450050870183525646, 11.20009191630321018153376520163, 12.40630134502689394687409331838, 12.84160533280932639723836506271