| L(s) = 1 | + (−1.31 − 0.520i)2-s + (−1.29 + 0.324i)3-s + (1.45 + 1.36i)4-s + (−0.836 − 0.395i)5-s + (1.87 + 0.247i)6-s + (−0.263 − 0.867i)7-s + (−1.20 − 2.55i)8-s + (−1.07 + 0.573i)9-s + (0.894 + 0.955i)10-s + (6.27 − 0.931i)11-s + (−2.33 − 1.29i)12-s + (−1.67 + 4.68i)13-s + (−0.105 + 1.27i)14-s + (1.21 + 0.241i)15-s + (0.254 + 3.99i)16-s + (3.44 − 0.685i)17-s + ⋯ |
| L(s) = 1 | + (−0.929 − 0.367i)2-s + (−0.747 + 0.187i)3-s + (0.729 + 0.684i)4-s + (−0.374 − 0.176i)5-s + (0.764 + 0.100i)6-s + (−0.0994 − 0.327i)7-s + (−0.426 − 0.904i)8-s + (−0.357 + 0.191i)9-s + (0.282 + 0.302i)10-s + (1.89 − 0.280i)11-s + (−0.673 − 0.375i)12-s + (−0.465 + 1.29i)13-s + (−0.0281 + 0.341i)14-s + (0.312 + 0.0622i)15-s + (0.0636 + 0.997i)16-s + (0.835 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.629421 + 0.0479296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629421 + 0.0479296i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 + 0.520i)T \) |
| good | 3 | \( 1 + (1.29 - 0.324i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (0.836 + 0.395i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.263 + 0.867i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-6.27 + 0.931i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (1.67 - 4.68i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-3.44 + 0.685i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-2.63 - 2.38i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-7.02 + 0.692i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-4.93 - 6.65i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-3.49 - 1.44i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.0343 + 0.699i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (0.225 - 0.274i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (2.18 - 8.72i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (10.9 + 7.32i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (4.46 - 6.01i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (1.61 + 4.51i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (3.33 - 5.56i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-6.51 - 3.90i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (1.30 - 2.43i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 8.45i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.31 - 1.97i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.578 + 11.7i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-1.62 - 0.160i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (4.64 - 11.2i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.78909248222126408147544452243, −11.25640777344743131699373759404, −10.11755442849396609107248350522, −9.242191392014591169178423386741, −8.372205792553783758101021458801, −7.04257118653911315614813474134, −6.32631682819942430983720355967, −4.64022300733041061892209141775, −3.33422169028160255318944528107, −1.21407920028719404902017559987,
0.946815419935210780916378844191, 3.13497117368198656788665615716, 5.16663845257279390211151553624, 6.17580686953661197349393313322, 6.99435875236479639022287962089, 8.052595463175662967614299836756, 9.170010702391981659460893409370, 9.930824953936077644760462276421, 11.18407763516322408360667117559, 11.72971807236035096490019876023
