L(s) = 1 | + (−1.58 + 0.696i)3-s + (−1.99 − 1.00i)5-s + (0.814 + 0.814i)7-s + (2.03 − 2.20i)9-s + (3.46 − 1.12i)11-s + (−2.63 − 5.16i)13-s + (3.86 + 0.207i)15-s + (−0.902 + 0.142i)17-s + (4.29 − 5.91i)19-s + (−1.85 − 0.724i)21-s + (3.05 − 6.00i)23-s + (2.97 + 4.02i)25-s + (−1.68 + 4.91i)27-s + (−4.30 + 3.12i)29-s + (−1.78 − 1.29i)31-s + ⋯ |
L(s) = 1 | + (−0.915 + 0.401i)3-s + (−0.892 − 0.450i)5-s + (0.307 + 0.307i)7-s + (0.677 − 0.735i)9-s + (1.04 − 0.339i)11-s + (−0.730 − 1.43i)13-s + (0.998 + 0.0536i)15-s + (−0.218 + 0.0346i)17-s + (0.986 − 1.35i)19-s + (−0.405 − 0.158i)21-s + (0.637 − 1.25i)23-s + (0.594 + 0.804i)25-s + (−0.324 + 0.945i)27-s + (−0.799 + 0.580i)29-s + (−0.321 − 0.233i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.662280 - 0.399890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.662280 - 0.399890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 - 0.696i)T \) |
| 5 | \( 1 + (1.99 + 1.00i)T \) |
good | 7 | \( 1 + (-0.814 - 0.814i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.46 + 1.12i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.63 + 5.16i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.902 - 0.142i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.29 + 5.91i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.05 + 6.00i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (4.30 - 3.12i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.78 + 1.29i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.74 + 1.39i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.66 - 0.866i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (3.01 - 3.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.998 - 6.30i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (8.04 + 1.27i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.98 + 6.10i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.21 + 12.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.976 - 6.16i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-5.88 - 8.09i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.26 + 4.21i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-4.29 - 5.90i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.190 + 1.20i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.41 - 10.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.84 - 0.609i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.35447576377901600747831970515, −11.07048279823676847699474451253, −9.682531317146189565322680357759, −8.849274717043607195890977956432, −7.67165995095191590299128011410, −6.64795099052132928003210021735, −5.31722375934568851230747700010, −4.62617835156751335588236982302, −3.28143688865912131567043954467, −0.69755398729664463115506830766,
1.61227307357206929733941507787, 3.78352098559587992586089968469, 4.73158937565011179222709837254, 6.12051207697305246887222402969, 7.22001800244153980465917768856, 7.57123083560437030844662797392, 9.182216570701454358330361880349, 10.17860771559951253339305485285, 11.44581241760197592083336229349, 11.63903065234850920046979477506