L(s) = 1 | + (−0.866 − 0.5i)2-s + (−2.29 − 1.32i)3-s + (0.499 + 0.866i)4-s + (1.32 + 2.29i)6-s − 1.64i·7-s − 0.999i·8-s + (2 + 3.46i)9-s + 0.645·11-s − 2.64i·12-s + (−1.73 + i)13-s + (−0.822 + 1.42i)14-s + (−0.5 + 0.866i)16-s − 3.99i·18-s + (−4.32 − 0.559i)19-s + (−2.17 + 3.77i)21-s + (−0.559 − 0.322i)22-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−1.32 − 0.763i)3-s + (0.249 + 0.433i)4-s + (0.540 + 0.935i)6-s − 0.622i·7-s − 0.353i·8-s + (0.666 + 1.15i)9-s + 0.194·11-s − 0.763i·12-s + (−0.480 + 0.277i)13-s + (−0.219 + 0.380i)14-s + (−0.125 + 0.216i)16-s − 0.942i·18-s + (−0.991 − 0.128i)19-s + (−0.475 + 0.822i)21-s + (−0.119 − 0.0688i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.312667 + 0.108452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.312667 + 0.108452i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.32 + 0.559i)T \) |
good | 3 | \( 1 + (2.29 + 1.32i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 1.64iT - 7T^{2} \) |
| 11 | \( 1 - 0.645T + 11T^{2} \) |
| 13 | \( 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.15 - 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.82 + 3.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.354T + 31T^{2} \) |
| 37 | \( 1 - 5.64iT - 37T^{2} \) |
| 41 | \( 1 + (5.14 - 8.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.613 - 0.354i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.35 + 4.82i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.43 - 4.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.46 - 12.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.02 - 2.32i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.64 + 11.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 6.14i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.93iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 7.14i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.30690426943898531009223850361, −9.508211117113618973720155303643, −8.341009125192234061993965461475, −7.49150350378548285376549541219, −6.75293315051653733577390904962, −6.09968511790868943036296631082, −4.94614655867458226934207474644, −3.88330662090817655171408159523, −2.21859957061049400579447179286, −1.02680834663582681728263043362,
0.27094206238253439348475482884, 2.17042949538327674722604005457, 3.89523923551544531744140535943, 4.95962137165043507300076379789, 5.68276201890826384682121853311, 6.35533752652899281660898408203, 7.30775880952663028928190890667, 8.448397361978245024234975092574, 9.213506032726972522023253848679, 10.07275226093903943263923706571