L(s) = 1 | + (0.933 + 1.45i)3-s + (0.313 + 0.313i)5-s + (0.0745 − 0.278i)7-s + (−1.25 + 2.72i)9-s + (−0.150 − 0.563i)11-s + (−1.79 + 3.12i)13-s + (−0.164 + 0.749i)15-s + (2.79 + 4.84i)17-s + (6.79 + 1.81i)19-s + (0.475 − 0.151i)21-s + (−3.32 + 5.76i)23-s − 4.80i·25-s + (−5.14 + 0.713i)27-s + (−3.57 − 2.06i)29-s + (1.03 − 1.03i)31-s + ⋯ |
L(s) = 1 | + (0.539 + 0.842i)3-s + (0.140 + 0.140i)5-s + (0.0281 − 0.105i)7-s + (−0.418 + 0.908i)9-s + (−0.0454 − 0.169i)11-s + (−0.496 + 0.867i)13-s + (−0.0424 + 0.193i)15-s + (0.678 + 1.17i)17-s + (1.55 + 0.417i)19-s + (0.103 − 0.0329i)21-s + (−0.693 + 1.20i)23-s − 0.960i·25-s + (−0.990 + 0.137i)27-s + (−0.664 − 0.383i)29-s + (0.186 − 0.186i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0109 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0109 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20058 + 1.21380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20058 + 1.21380i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.933 - 1.45i)T \) |
| 13 | \( 1 + (1.79 - 3.12i)T \) |
good | 5 | \( 1 + (-0.313 - 0.313i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.0745 + 0.278i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.150 + 0.563i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.79 - 4.84i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.79 - 1.81i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.32 - 5.76i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.57 + 2.06i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.03 + 1.03i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.72 + 1.80i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (7.36 - 1.97i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.26 + 1.88i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.71 - 3.71i)T - 47iT^{2} \) |
| 53 | \( 1 + 3.64iT - 53T^{2} \) |
| 59 | \( 1 + (-3.29 - 0.881i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (5.25 + 9.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.29 + 8.55i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 14.8i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.52 - 3.52i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 + (-8.23 - 8.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.64 - 13.6i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.59 - 0.694i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.67505215166403582646767129704, −9.726919659473336115706569497079, −9.442594635405277120529563939685, −8.121546533793994200549309152559, −7.61459907058132026122462407782, −6.17907025726815877196389612016, −5.26870639481386701076177918537, −4.12428514603863697750219341251, −3.29516726006715121319244216472, −1.90968569861151687676764345781,
0.926853697972400186437514812271, 2.49198429777431768334851869650, 3.36705190326652543106625629806, 5.00246787068216264307535036700, 5.84023766715523044801848547786, 7.15805408381413355491062516750, 7.56760797943175763606672037523, 8.597202727508270756310775758597, 9.467960938780041250939888658449, 10.17697001192881913034278863346