L(s) = 1 | + (1.36 + 2.36i)2-s + (−0.5 − 0.866i)3-s + (−2.73 + 4.73i)4-s + 5-s + (1.36 − 2.36i)6-s + (−0.866 + 1.5i)7-s − 9.46·8-s + (−0.499 + 0.866i)9-s + (1.36 + 2.36i)10-s + (1 + 1.73i)11-s + 5.46·12-s + (3.59 − 0.232i)13-s − 4.73·14-s + (−0.5 − 0.866i)15-s + (−7.46 − 12.9i)16-s + (1.36 − 2.36i)17-s + ⋯ |
L(s) = 1 | + (0.965 + 1.67i)2-s + (−0.288 − 0.499i)3-s + (−1.36 + 2.36i)4-s + 0.447·5-s + (0.557 − 0.965i)6-s + (−0.327 + 0.566i)7-s − 3.34·8-s + (−0.166 + 0.288i)9-s + (0.431 + 0.748i)10-s + (0.301 + 0.522i)11-s + 1.57·12-s + (0.997 − 0.0643i)13-s − 1.26·14-s + (−0.129 − 0.223i)15-s + (−1.86 − 3.23i)16-s + (0.331 − 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.572191 + 1.53273i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.572191 + 1.53273i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-3.59 + 0.232i)T \) |
good | 2 | \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (0.866 - 1.5i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.36 + 2.36i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.73 + 6.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.36 - 5.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + (5.19 + 9i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.63 - 6.29i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.598 - 1.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + (2.83 - 4.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.69 + 13.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.598 + 1.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.633 + 1.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + (4.36 + 7.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.59 + 4.5i)T + (-48.5 - 84.0i)T^{2} \) |
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
−13.04243327971250475792839699374, −12.47882981537166903256519954048, −11.37109497535048415253955850421, −9.404743442113346322941552398547, −8.593937219206852006091147951902, −7.33144503469277607537118619975, −6.59667619928395449414703329936, −5.68759122044395370545909048966, −4.77746623556523935925062039256, −3.11580129395178037719358741179,
1.38176906633482476661311864734, 3.33032416961313508940299034771, 4.02901896840208229107736067188, 5.47417495401766186800803169318, 6.22346768117121403045275735169, 8.603390932594624818858022422659, 9.921627800006830445413681924986, 10.21438197138394544086701684743, 11.28752560995680720658426065989, 11.98162997421715015239795622739