| L(s) = 1 | + (1 − i)3-s + (−1 + i)7-s + i·9-s − 4·11-s + (−3 − 3i)13-s + (3 + 3i)17-s + 6i·19-s + 2i·21-s + (−3 − 3i)23-s + (4 + 4i)27-s − 2·29-s + 6i·31-s + (−4 + 4i)33-s + (−3 + 3i)37-s − 6·39-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.377 + 0.377i)7-s + 0.333i·9-s − 1.20·11-s + (−0.832 − 0.832i)13-s + (0.727 + 0.727i)17-s + 1.37i·19-s + 0.436i·21-s + (−0.625 − 0.625i)23-s + (0.769 + 0.769i)27-s − 0.371·29-s + 1.07i·31-s + (−0.696 + 0.696i)33-s + (−0.493 + 0.493i)37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9491331485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9491331485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 23iT^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (9 - 9i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + (9 + 9i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (3 - 3i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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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
−9.766610285977486184781536614746, −8.567861705820082026402441543274, −7.86472366519306361174980449743, −7.67582681400150810228972534377, −6.39474630611944680003834418111, −5.57794756560665241606885970344, −4.79556262901023302769902494080, −3.34824530075107269887668813136, −2.66418800632813597259367022339, −1.62850274420999023814527468902,
0.31641206405672648544789529484, 2.28193055021139060301086659347, 3.11186415975879097431111809649, 4.06173416028383938755914560932, 4.91208335368472627602154805259, 5.80522194710482539098642063165, 7.06202566632976138060279908020, 7.44244386212684433193993942438, 8.557942136292391511888144813265, 9.285152345849520618027460352812
