L(s) = 1 | + (−0.456 + 0.456i)3-s + (−0.456 + 2.18i)5-s + (2.79 + 2.79i)7-s + 2.58i·9-s + 4.37i·11-s + (1.73 + 1.73i)13-s + (−0.791 − 1.20i)15-s + (3 − 3i)17-s − 3.46·19-s − 2.55·21-s + (0.791 − 0.791i)23-s + (−4.58 − 1.99i)25-s + (−2.55 − 2.55i)27-s − 5.29i·29-s − 1.58i·31-s + ⋯ |
L(s) = 1 | + (−0.263 + 0.263i)3-s + (−0.204 + 0.978i)5-s + (1.05 + 1.05i)7-s + 0.860i·9-s + 1.31i·11-s + (0.480 + 0.480i)13-s + (−0.204 − 0.312i)15-s + (0.727 − 0.727i)17-s − 0.794·19-s − 0.556·21-s + (0.164 − 0.164i)23-s + (−0.916 − 0.399i)25-s + (−0.490 − 0.490i)27-s − 0.982i·29-s − 0.284i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.547563321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547563321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.456 - 2.18i)T \) |
good | 3 | \( 1 + (0.456 - 0.456i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.37iT - 11T^{2} \) |
| 13 | \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + (-0.791 + 0.791i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.29iT - 29T^{2} \) |
| 31 | \( 1 + 1.58iT - 31T^{2} \) |
| 37 | \( 1 + (-5.19 + 5.19i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 + (8.29 - 8.29i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.791 - 0.791i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.64 + 2.64i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 + (-8.29 - 8.29i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (0.582 + 0.582i)T + 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (4.83 - 4.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.16iT - 89T^{2} \) |
| 97 | \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \) |
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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
−9.980170227962879890197577532146, −9.326592588784720746482302793811, −8.093915224318593131964328124981, −7.72047845281256963873986362154, −6.67127813840145706316822659215, −5.75140980514739356819604182328, −4.85374972059535829475138336660, −4.14447391226595743705326980124, −2.57880174735151263626083675754, −1.95661921150556404411177555339,
0.73863947166098261689474477983, 1.41123003826392872923080840141, 3.43496160377158102103579571441, 4.10166165593809281022061175808, 5.19189921513966913484627391878, 5.93069106654349255103832075647, 6.89027379623566375728749711115, 8.032871359000774109805535726754, 8.326487978305407173907474873814, 9.190152397866047156281023722675