L(s) = 1 | + (0.296 + 0.512i)2-s + (−0.866 + 0.5i)3-s + (0.824 − 1.42i)4-s + (−0.903 − 2.04i)5-s + (−0.512 − 0.296i)6-s + (0.828 − 1.43i)7-s + 2.16·8-s + (0.499 − 0.866i)9-s + (0.781 − 1.06i)10-s + (−1.22 + 0.706i)11-s + 1.64i·12-s + (0.0145 − 3.60i)13-s + 0.981·14-s + (1.80 + 1.31i)15-s + (−1.00 − 1.74i)16-s + (2.83 + 1.63i)17-s + ⋯ |
L(s) = 1 | + (0.209 + 0.362i)2-s + (−0.499 + 0.288i)3-s + (0.412 − 0.714i)4-s + (−0.404 − 0.914i)5-s + (−0.209 − 0.120i)6-s + (0.313 − 0.542i)7-s + 0.763·8-s + (0.166 − 0.288i)9-s + (0.247 − 0.337i)10-s + (−0.368 + 0.212i)11-s + 0.476i·12-s + (0.00404 − 0.999i)13-s + 0.262·14-s + (0.466 + 0.340i)15-s + (−0.252 − 0.437i)16-s + (0.688 + 0.397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14273 - 0.383852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14273 - 0.383852i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.903 + 2.04i)T \) |
| 13 | \( 1 + (-0.0145 + 3.60i)T \) |
good | 2 | \( 1 + (-0.296 - 0.512i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.828 + 1.43i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.706i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.83 - 1.63i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.29 - 4.21i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.70 - 3.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.252 - 0.438i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.791iT - 31T^{2} \) |
| 37 | \( 1 + (2.37 + 4.12i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.67 - 4.42i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.93 + 1.11i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.43T + 47T^{2} \) |
| 53 | \( 1 - 0.422iT - 53T^{2} \) |
| 59 | \( 1 + (-11.3 - 6.54i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.463 - 0.803i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.21 - 10.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.947 - 0.547i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 - 4.02T + 83T^{2} \) |
| 89 | \( 1 + (0.517 - 0.299i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.57 - 2.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.27231253214591688496165889248, −11.50487427315609350497936938863, −10.31986072070940199733086378708, −9.794098182806564528478496385853, −8.069606824846718862409392424853, −7.36418127485420416864852439600, −5.69086228015661493394144942907, −5.24373736611301200728878252844, −3.84972283712217462816462086561, −1.21850054942205504329490792439,
2.30087746504972072311122588765, 3.54404831788231977761223664846, 5.07531862184227454539987939440, 6.58777947861440651721356251022, 7.38078719952077664402653503414, 8.348412153069201697647995942923, 9.933898774111487294204104837281, 11.04875043908584939033515247322, 11.80231653633297785568349778526, 12.10173964788180538110669356723