| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.959 + 0.281i)6-s + (4.30 + 2.76i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (0.877 + 6.10i)11-s + (0.142 + 0.989i)12-s + (−1.50 + 0.967i)13-s + (3.34 − 3.86i)14-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (−1.59 + 0.468i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−0.391 + 0.115i)6-s + (1.62 + 1.04i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.266 + 0.170i)10-s + (0.264 + 1.84i)11-s + (0.0410 + 0.285i)12-s + (−0.417 + 0.268i)13-s + (0.895 − 1.03i)14-s + (−0.107 + 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.386 + 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45760 - 0.144604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45760 - 0.144604i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.76 - 4.45i)T \) |
| good | 7 | \( 1 + (-4.30 - 2.76i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.877 - 6.10i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.50 - 0.967i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.59 - 0.468i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-6.35 - 1.86i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (4.10 - 1.20i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.672 - 1.47i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.51 + 5.20i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (5.47 + 6.31i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (5.39 + 11.8i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + (-3.12 - 2.00i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.44 - 3.50i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.76 + 6.05i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.166 - 1.15i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.910 + 6.33i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.56 - 1.04i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.73 - 3.03i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.22 + 4.87i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.04 - 2.28i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.42 - 3.95i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.63831401835796692882207400971, −9.503909414444638647227979531177, −8.868658472352994648757565871522, −7.75119689431081362197253368200, −7.24105371021856741482227710725, −5.50178832637427631714457260650, −5.05242553067967448861608272367, −4.00449645436042949540124885719, −2.17415701364764494666104968184, −1.62094060528447923285606473588,
0.856567860404744621585609249379, 3.15243492228200526875738293104, 4.23486214477877550713321094097, 4.99799537659598607830649970535, 5.95045781264832071277018250739, 7.04869678540349687598679944153, 7.962453003587465508678184124896, 8.437891268264521646918273525755, 9.626280274294834092280011389210, 10.63156411501736276816398681452
