| L(s) = 1 | + (0.0173 − 1.41i)2-s + (1.65 − 0.830i)3-s + (−1.99 − 0.0490i)4-s + (0.815 + 1.83i)5-s + (−1.14 − 2.34i)6-s + (1.37 + 0.821i)7-s + (−0.103 + 2.82i)8-s + (0.245 − 0.331i)9-s + (2.61 − 1.12i)10-s + (5.16 + 4.03i)11-s + (−3.33 + 1.57i)12-s + (1.84 − 1.75i)13-s + (1.18 − 1.92i)14-s + (2.87 + 2.35i)15-s + (3.99 + 0.196i)16-s + (−1.30 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (0.0122 − 0.999i)2-s + (0.952 − 0.479i)3-s + (−0.999 − 0.0245i)4-s + (0.364 + 0.822i)5-s + (−0.467 − 0.958i)6-s + (0.518 + 0.310i)7-s + (−0.0367 + 0.999i)8-s + (0.0819 − 0.110i)9-s + (0.827 − 0.354i)10-s + (1.55 + 1.21i)11-s + (−0.964 + 0.456i)12-s + (0.511 − 0.486i)13-s + (0.316 − 0.514i)14-s + (0.741 + 0.608i)15-s + (0.998 + 0.0490i)16-s + (−0.316 − 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82613 - 0.891139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82613 - 0.891139i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0173 + 1.41i)T \) |
| good | 3 | \( 1 + (-1.65 + 0.830i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.815 - 1.83i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-1.37 - 0.821i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-5.16 - 4.03i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 1.75i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.30 + 1.58i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.602 + 3.47i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (5.10 + 2.41i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (4.47 + 2.53i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (7.49 - 1.49i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-5.24 - 3.32i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-4.93 - 5.44i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (0.749 + 2.26i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (0.319 + 0.170i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (4.28 - 2.42i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-8.34 + 8.76i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.173 + 2.34i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (6.84 + 5.90i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (6.17 + 0.916i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-2.30 + 1.37i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-6.25 - 1.89i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-1.98 + 1.25i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 4.84i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (6.95 + 4.64i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.91116743903422716939282126087, −9.773789559028255987972451996573, −9.155019398433822103970690356686, −8.295203855964444305046826539289, −7.31973756563737699734633444510, −6.21647617802016544586037969001, −4.74638878788216271206250740452, −3.62498490027990516779937883708, −2.45148096360379525201524918823, −1.74249031311867558792946196841,
1.40125900120095929303751588884, 3.80004999223109603611025616536, 4.04040001960710735492542099587, 5.60229612180990052477121711541, 6.27536484311402558826622612249, 7.64385950825867038961436793698, 8.529974046596306371583099994852, 9.064804209406689999194152148455, 9.506804892030960594578223821150, 10.90169709649445711493075977034
