| L(s) = 1 | + (0.382 + 1.36i)2-s + (2.45 + 0.214i)3-s + (−1.70 + 1.04i)4-s + (0.179 − 0.0836i)5-s + (0.646 + 3.42i)6-s + (3.23 + 1.86i)7-s + (−2.07 − 1.92i)8-s + (3.01 + 0.530i)9-s + (0.182 + 0.212i)10-s + (−3.07 − 0.824i)11-s + (−4.40 + 2.18i)12-s + (−0.457 − 5.22i)13-s + (−1.30 + 5.11i)14-s + (0.457 − 0.166i)15-s + (1.82 − 3.55i)16-s + (0.0431 + 0.244i)17-s + ⋯ |
| L(s) = 1 | + (0.270 + 0.962i)2-s + (1.41 + 0.123i)3-s + (−0.853 + 0.521i)4-s + (0.0802 − 0.0374i)5-s + (0.263 + 1.39i)6-s + (1.22 + 0.705i)7-s + (−0.732 − 0.680i)8-s + (1.00 + 0.176i)9-s + (0.0577 + 0.0670i)10-s + (−0.927 − 0.248i)11-s + (−1.27 + 0.632i)12-s + (−0.126 − 1.44i)13-s + (−0.348 + 1.36i)14-s + (0.118 − 0.0430i)15-s + (0.456 − 0.889i)16-s + (0.0104 + 0.0593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0892 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0892 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58889 + 1.45283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58889 + 1.45283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.382 - 1.36i)T \) |
| 19 | \( 1 + (4.30 - 0.655i)T \) |
| good | 3 | \( 1 + (-2.45 - 0.214i)T + (2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (-0.179 + 0.0836i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-3.23 - 1.86i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.07 + 0.824i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.457 + 5.22i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.0431 - 0.244i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.10 - 5.78i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.258 + 0.181i)T + (9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (-4.31 + 7.47i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.668 + 0.668i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.15 + 7.34i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.74 + 2.68i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.885 - 5.02i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.49 - 5.35i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (2.87 + 2.01i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-0.849 - 0.395i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-4.35 + 3.04i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-3.46 + 9.52i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.47 - 4.13i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 9.59i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.10 + 0.564i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (10.7 - 12.7i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.57 - 8.92i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.29020323095897349896123342359, −10.90740058517899347893262576332, −9.641930265598833777727887542225, −8.758664901800597543906704055926, −7.88295114643548172217620193056, −7.76065714620381750539772899488, −5.82124869936215196124078281111, −5.02766781276982168686852565909, −3.62960697691416756380285206470, −2.42159743867434001770646087503,
1.76840476925194067104800445717, 2.67622091750255598092489166286, 4.17077412692628936550525663833, 4.82102005066765974128963315536, 6.78159203408550288719601221301, 8.166698350392216768303569117372, 8.549291025444268664088821311537, 9.717661432404749174185615314663, 10.57257565743638101774188981238, 11.42990116221299079215691103736
