| L(s) = 1 | + (1.34 + 0.426i)2-s + (1.96 − 2.39i)3-s + (1.63 + 1.14i)4-s + (−0.956 + 0.290i)5-s + (3.67 − 2.39i)6-s + (4.41 + 0.877i)7-s + (1.71 + 2.24i)8-s + (−1.28 − 6.48i)9-s + (−1.41 − 0.0165i)10-s + (−3.98 − 0.392i)11-s + (5.97 − 1.66i)12-s + (−1.64 + 5.42i)13-s + (5.57 + 3.06i)14-s + (−1.18 + 2.86i)15-s + (1.35 + 3.76i)16-s + (−1.47 − 3.55i)17-s + ⋯ |
| L(s) = 1 | + (0.953 + 0.301i)2-s + (1.13 − 1.38i)3-s + (0.818 + 0.574i)4-s + (−0.427 + 0.129i)5-s + (1.49 − 0.976i)6-s + (1.66 + 0.331i)7-s + (0.606 + 0.794i)8-s + (−0.429 − 2.16i)9-s + (−0.447 − 0.00522i)10-s + (−1.20 − 0.118i)11-s + (1.72 − 0.479i)12-s + (−0.456 + 1.50i)13-s + (1.49 + 0.819i)14-s + (−0.306 + 0.739i)15-s + (0.339 + 0.940i)16-s + (−0.356 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.63889 - 0.785971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.63889 - 0.785971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 - 0.426i)T \) |
| 5 | \( 1 + (0.956 - 0.290i)T \) |
| good | 3 | \( 1 + (-1.96 + 2.39i)T + (-0.585 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-4.41 - 0.877i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (3.98 + 0.392i)T + (10.7 + 2.14i)T^{2} \) |
| 13 | \( 1 + (1.64 - 5.42i)T + (-10.8 - 7.22i)T^{2} \) |
| 17 | \( 1 + (1.47 + 3.55i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (2.49 - 1.33i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (2.67 - 4.00i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.771 + 7.83i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (5.27 + 5.27i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.184 + 0.344i)T + (-20.5 - 30.7i)T^{2} \) |
| 41 | \( 1 + (1.69 + 1.13i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-2.05 - 2.50i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-2.54 + 1.05i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.00307 - 0.0312i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (0.326 + 1.07i)T + (-49.0 + 32.7i)T^{2} \) |
| 61 | \( 1 + (-2.69 - 2.21i)T + (11.9 + 59.8i)T^{2} \) |
| 67 | \( 1 + (5.60 + 4.59i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (-1.81 + 9.10i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-4.08 + 0.812i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-8.94 - 3.70i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-4.40 - 8.23i)T + (-46.1 + 69.0i)T^{2} \) |
| 89 | \( 1 + (3.03 + 4.54i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 10.4i)T + 97iT^{2} \) |
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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
−11.02824876850674605930201202753, −9.259180300037083301124180387692, −8.240880354427443788640453892361, −7.73066954987347770514563582781, −7.26649652072935705675392330678, −6.11929200341268695653558043068, −4.92446351442325104716170819356, −3.89885664544837042372742861306, −2.36384838747364544171983404531, −2.00507338185578153062343210411,
2.08670711874466778184126563266, 3.11331796149210371608321167598, 4.13193038459564518554262000234, 4.89565179217642091130415056278, 5.37893377352838493813728637641, 7.41206160689251561071063139024, 8.102600014977673766327875894809, 8.739297599542304847549654443311, 10.33134068309874818502555300568, 10.53144303835844239895723957644
