| L(s) = 1 | + (−1 + i)3-s + i·7-s + i·9-s + (2 + 2i)11-s + (−2 + 2i)13-s − 6·17-s + (3 − 3i)19-s + (−1 − i)21-s − 5i·25-s + (−4 − 4i)27-s + (−1 + i)29-s − 10·31-s − 4·33-s + (3 + 3i)37-s − 4i·39-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + 0.377i·7-s + 0.333i·9-s + (0.603 + 0.603i)11-s + (−0.554 + 0.554i)13-s − 1.45·17-s + (0.688 − 0.688i)19-s + (−0.218 − 0.218i)21-s − i·25-s + (−0.769 − 0.769i)27-s + (−0.185 + 0.185i)29-s − 1.79·31-s − 0.696·33-s + (0.493 + 0.493i)37-s − 0.640i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 5 | \( 1 + 5iT^{2} \) |
| 11 | \( 1 + (-2 - 2i)T + 11iT^{2} \) |
| 13 | \( 1 + (2 - 2i)T - 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-3 + 3i)T - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (1 - i)T - 29iT^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + (6 + 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1 - i)T + 59iT^{2} \) |
| 61 | \( 1 - 61iT^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.440287009060625157677459105039, −7.46210067843467495822047838550, −6.79396448796361483714131232033, −6.08267192462224192712747291132, −5.06293670951639563209588926909, −4.66468158522138578700967490351, −3.88288141314023248493577835272, −2.58257068498875587442373205522, −1.78945735514046629016215899328, 0,
1.09082795927623013946800225721, 2.14490216840986210925220828057, 3.47070478609745531333952863288, 4.00746153818772332526731938225, 5.33238298363205280371230262034, 5.70277842224657234115772809869, 6.77535871634740086493230651392, 7.05830537809629821034437017401, 7.917938796026827759911746443961
