| L(s) = 1 | + (−1.39 + 0.252i)2-s + (−1.88 + 1.00i)3-s + (1.87 − 0.702i)4-s + (−0.634 − 0.773i)5-s + (2.36 − 1.87i)6-s + (−0.780 + 0.521i)7-s + (−2.42 + 1.45i)8-s + (0.872 − 1.30i)9-s + (1.07 + 0.915i)10-s + (4.10 − 1.24i)11-s + (−2.82 + 3.21i)12-s + (−3.53 − 2.90i)13-s + (0.953 − 0.922i)14-s + (1.97 + 0.818i)15-s + (3.01 − 2.63i)16-s + (−1.91 + 0.794i)17-s + ⋯ |
| L(s) = 1 | + (−0.983 + 0.178i)2-s + (−1.08 + 0.581i)3-s + (0.936 − 0.351i)4-s + (−0.283 − 0.345i)5-s + (0.967 − 0.766i)6-s + (−0.294 + 0.196i)7-s + (−0.858 + 0.513i)8-s + (0.290 − 0.435i)9-s + (0.340 + 0.289i)10-s + (1.23 − 0.375i)11-s + (−0.814 + 0.927i)12-s + (−0.980 − 0.804i)13-s + (0.254 − 0.246i)14-s + (0.509 + 0.211i)15-s + (0.752 − 0.658i)16-s + (−0.465 + 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.373553 + 0.316017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373553 + 0.316017i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.252i)T \) |
| 5 | \( 1 + (0.634 + 0.773i)T \) |
| good | 3 | \( 1 + (1.88 - 1.00i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (0.780 - 0.521i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.10 + 1.24i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (3.53 + 2.90i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.91 - 0.794i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.281 - 2.86i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-9.21 + 1.83i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.848 - 2.79i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (2.47 + 2.47i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.881 - 0.0868i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 9.93i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.764 - 0.408i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.73 - 6.60i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.44 - 11.3i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (4.78 - 3.92i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (6.43 + 12.0i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-4.75 - 8.89i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-6.36 - 9.53i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-4.72 - 3.15i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (5.04 - 12.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-4.23 + 0.416i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-2.50 - 0.497i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-2.08 - 2.08i)T + 97iT^{2} \) |
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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
−10.95340465487944872343860299824, −9.783025259104132081038112302155, −9.242341797416619186699669720396, −8.274190686202602392969106223416, −7.21896900910956312378023399824, −6.27572327355130433191798399043, −5.51714079878167996718139263863, −4.48151892721243889891473243205, −2.93841181475864805678994472142, −0.997178758299031097561608472466,
0.54531054484213931881445592273, 2.01226043059760878130260070160, 3.53283775374600480443254866894, 4.99014355830118967609091744281, 6.36947239926935195098833651850, 7.00546777912365111343815567922, 7.29449553391215886931004933984, 8.959784555990695606639678253357, 9.373018884230030849786726219879, 10.54696785258371885514619900306
