| 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
−9.285152345849520618027460352812, −8.557942136292391511888144813265, −7.44244386212684433193993942438, −7.06202566632976138060279908020, −5.80522194710482539098642063165, −4.91208335368472627602154805259, −4.06173416028383938755914560932, −3.11186415975879097431111809649, −2.28193055021139060301086659347, −0.31641206405672648544789529484,
1.62850274420999023814527468902, 2.66418800632813597259367022339, 3.34824530075107269887668813136, 4.79556262901023302769902494080, 5.57794756560665241606885970344, 6.39474630611944680003834418111, 7.67582681400150810228972534377, 7.86472366519306361174980449743, 8.567861705820082026402441543274, 9.766610285977486184781536614746
