L(s) = 1 | + (−0.579 + 1.00i)3-s + (−0.5 + 0.866i)5-s + 2.43·7-s + (0.827 + 1.43i)9-s + 5.75·11-s + (0.797 + 1.38i)13-s + (−0.579 − 1.00i)15-s + (2.99 − 5.18i)17-s + (−0.149 − 4.35i)19-s + (−1.41 + 2.44i)21-s + (−0.470 − 0.814i)23-s + (−0.499 − 0.866i)25-s − 5.39·27-s + (−1.30 − 2.26i)29-s + 5.26·31-s + ⋯ |
L(s) = 1 | + (−0.334 + 0.579i)3-s + (−0.223 + 0.387i)5-s + 0.920·7-s + (0.275 + 0.477i)9-s + 1.73·11-s + (0.221 + 0.383i)13-s + (−0.149 − 0.259i)15-s + (0.725 − 1.25i)17-s + (−0.0342 − 0.999i)19-s + (−0.308 + 0.533i)21-s + (−0.0980 − 0.169i)23-s + (−0.0999 − 0.173i)25-s − 1.03·27-s + (−0.243 − 0.421i)29-s + 0.946·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930464451\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930464451\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.149 + 4.35i)T \) |
good | 3 | \( 1 + (0.579 - 1.00i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 13 | \( 1 + (-0.797 - 1.38i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.99 + 5.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.470 + 0.814i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.26 + 3.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.47 - 7.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.09 - 1.90i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.39 - 9.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 - 9.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.504 - 0.874i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.41 + 7.65i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.12 + 8.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.80 + 6.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.02 - 3.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.430224958216998144943908790697, −9.024743387594016696553207981485, −7.85683744270320385464156373297, −7.18996403278469297808215106182, −6.34300825897482744508740731502, −5.26560428278391707787453623604, −4.48646726715338130337315725456, −3.85001691503578083364876978780, −2.47548948774566073190943349525, −1.14819192795897110374669973161,
1.13822038520596102667790051142, 1.65415628602824726986864591345, 3.61350813700661900686264140956, 4.13737228118847901971674097789, 5.36338250197367988725250795977, 6.19753603109699213714181540446, 6.82876737261001827181880342032, 7.951067044822817914151759161145, 8.331634094200924403784137180799, 9.354175967593545959854728087799