| L(s) = 1 | + (−1 + i)3-s − 2·7-s + i·9-s + (−1 + i)11-s + (1 − i)13-s + 2i·17-s + (3 + 3i)19-s + (2 − 2i)21-s − 6·23-s + (−4 − 4i)27-s + (−3 − 3i)29-s + 8·31-s − 2i·33-s + (−3 − 3i)37-s + 2i·39-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s − 0.755·7-s + 0.333i·9-s + (−0.301 + 0.301i)11-s + (0.277 − 0.277i)13-s + 0.485i·17-s + (0.688 + 0.688i)19-s + (0.436 − 0.436i)21-s − 1.25·23-s + (−0.769 − 0.769i)27-s + (−0.557 − 0.557i)29-s + 1.43·31-s − 0.348i·33-s + (−0.493 − 0.493i)37-s + 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (5 + 5i)T + 43iT^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (3 - 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (9 + 9i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5 + 5i)T - 67iT^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (1 - i)T - 83iT^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.409849107021290924212913667598, −8.166221422442874923142527804299, −7.69637422811741651850934558385, −6.42877478120596479587860185234, −5.87631680476208146052773780967, −5.00401420807203998386668496800, −4.09648511318576568001315043004, −3.18816530161154160413273479350, −1.87310610378703042054930399652, 0,
1.28971217048944487230143332373, 2.77741016294620158090345512097, 3.66254060472490055928832640638, 4.86788396244498416072826967746, 5.82865531445608805589229308253, 6.47840116003058930629506805179, 7.10033155884986463762739968618, 8.021975407540844688834874989070, 8.973240765902173635808131042840
