L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s − 5-s + (1.5 − 0.866i)6-s + (0.5 − 2.59i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (0.866 + 0.5i)10-s + 4.73i·11-s − 1.73·12-s + (3 + 1.73i)13-s + (−1.73 + 2i)14-s + (0.866 − 1.5i)15-s + (−0.5 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.612 − 0.353i)6-s + (0.188 − 0.981i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.273 + 0.158i)10-s + 1.42i·11-s − 0.499·12-s + (0.832 + 0.480i)13-s + (−0.462 + 0.534i)14-s + (0.223 − 0.387i)15-s + (−0.125 + 0.216i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.223224 + 0.441612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.223224 + 0.441612i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.53iT - 23T^{2} \) |
| 29 | \( 1 + (5.59 - 3.23i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.09 - 4.09i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.09 + 5.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.59 + 2.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.29 + 3.63i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.09 - 1.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 6.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.73iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.29 - 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.59 - 9.69i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.19 + 3.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.39 + 4.26i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.97235769134560927274016142990, −10.03497049684643941034137921375, −9.426327412580283132929633999148, −8.461937319155074109746802266127, −7.35421664463322503674973931848, −6.68928564122393109300222088875, −5.21127362825560431148331972097, −4.17227448173198733089592572134, −3.52683880503321538642398586293, −1.56261958949668680484042128166,
0.35889196522443064424264991280, 1.94552673553787370584648428394, 3.37429495985151954643688675634, 5.23725999504448322484085722535, 5.90931070942197429662999659403, 6.65907050825912759500501845490, 7.85306146994668876273652100038, 8.393240649343308236261571448313, 9.050583950586621629954900932299, 10.49317487745292279000599243155