| L(s) = 1 | + (0.893 + 0.448i)2-s + (−0.977 − 1.42i)3-s + (0.597 + 0.802i)4-s + (−1.15 − 3.85i)5-s + (−0.232 − 1.71i)6-s + (−0.663 − 1.53i)7-s + (0.173 + 0.984i)8-s + (−1.08 + 2.79i)9-s + (0.698 − 3.95i)10-s + (0.0131 − 0.00311i)11-s + (0.562 − 1.63i)12-s + (3.84 + 2.52i)13-s + (0.0974 − 1.67i)14-s + (−4.37 + 5.41i)15-s + (−0.286 + 0.957i)16-s + (4.83 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (0.631 + 0.317i)2-s + (−0.564 − 0.825i)3-s + (0.298 + 0.401i)4-s + (−0.515 − 1.72i)5-s + (−0.0947 − 0.700i)6-s + (−0.250 − 0.581i)7-s + (0.0613 + 0.348i)8-s + (−0.362 + 0.931i)9-s + (0.220 − 1.25i)10-s + (0.00396 − 0.000940i)11-s + (0.162 − 0.472i)12-s + (1.06 + 0.700i)13-s + (0.0260 − 0.447i)14-s + (−1.13 + 1.39i)15-s + (−0.0717 + 0.239i)16-s + (1.17 − 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.998170 - 0.717448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.998170 - 0.717448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.893 - 0.448i)T \) |
| 3 | \( 1 + (0.977 + 1.42i)T \) |
| good | 5 | \( 1 + (1.15 + 3.85i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.663 + 1.53i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-0.0131 + 0.00311i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.84 - 2.52i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-4.83 + 1.76i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.66 - 0.968i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.253 - 0.586i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.262 + 4.51i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (9.48 - 1.10i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (-6.03 - 5.06i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-6.30 + 3.16i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-4.34 - 4.60i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (11.3 + 1.32i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (2.96 - 5.14i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.42 + 0.573i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-1.39 + 1.87i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.232 + 3.99i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (1.01 - 5.73i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.45 - 8.24i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (4.39 + 2.20i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-13.5 - 6.80i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-1.34 - 7.65i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.923 + 3.08i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−12.77212908430987473724468069901, −11.93668861421044934306911220819, −11.20683602756764017606046938236, −9.440312823929031638283700823856, −8.176472428141695464533694580017, −7.44335110572022236246041641847, −6.04247362970021325354621885187, −5.07863753516773752821957389157, −3.87666724418762481032841755459, −1.19906959414094080771113416884,
3.07235917575760619252547212669, 3.74950771546639171398364038439, 5.56509429996479597354524888586, 6.30674492376031348799511608126, 7.63826034680857031218645241644, 9.377584699715146685951199487983, 10.53283634674821549566968506546, 10.98257906057248118819904232103, 11.83546374463580783967421981404, 12.89167526446615321861551074309
