| L(s) = 1 | + (1.22 − 0.701i)2-s + (0.372 − 0.0656i)3-s + (1.01 − 1.72i)4-s + (−1.50 − 1.64i)5-s + (0.411 − 0.341i)6-s + (−0.391 + 0.678i)7-s + (0.0408 − 2.82i)8-s + (−2.68 + 0.977i)9-s + (−3.01 − 0.967i)10-s + (4.68 − 2.70i)11-s + (0.265 − 0.707i)12-s + (0.952 − 5.40i)13-s + (−0.00533 + 1.10i)14-s + (−0.670 − 0.515i)15-s + (−1.93 − 3.50i)16-s + (−0.973 + 2.67i)17-s + ⋯ |
| L(s) = 1 | + (0.868 − 0.495i)2-s + (0.214 − 0.0379i)3-s + (0.508 − 0.861i)4-s + (−0.675 − 0.737i)5-s + (0.167 − 0.139i)6-s + (−0.147 + 0.256i)7-s + (0.0144 − 0.999i)8-s + (−0.894 + 0.325i)9-s + (−0.952 − 0.305i)10-s + (1.41 − 0.816i)11-s + (0.0766 − 0.204i)12-s + (0.264 − 1.49i)13-s + (−0.00142 + 0.295i)14-s + (−0.173 − 0.132i)15-s + (−0.483 − 0.875i)16-s + (−0.236 + 0.648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37967 - 1.54553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37967 - 1.54553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.701i)T \) |
| 5 | \( 1 + (1.50 + 1.64i)T \) |
| 19 | \( 1 + (-4.05 - 1.59i)T \) |
| good | 3 | \( 1 + (-0.372 + 0.0656i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (0.391 - 0.678i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.68 + 2.70i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.952 + 5.40i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.973 - 2.67i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.999 - 0.838i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.22 - 6.10i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.09 - 7.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.57T + 37T^{2} \) |
| 41 | \( 1 + (-5.74 + 1.01i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.47 + 5.43i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.66 + 1.69i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.38 - 2.84i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.21 - 0.804i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (6.61 + 5.54i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.60 - 7.16i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.81 + 7.40i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (8.11 - 1.43i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.40 + 7.94i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.17 - 14.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.44 + 1.13i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.86 + 2.49i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.22048950135335513305480261164, −10.60259037740497271598073194353, −9.105227757504385515460344917755, −8.553051871482296236563201075969, −7.28334437761814040000291476138, −5.86483513932907802238131520223, −5.29138362430210044414547530026, −3.79727629825527013076069815601, −3.11978369934404908378609887708, −1.16142998136634385677426058405,
2.48711397027861070637837427912, 3.77299650144817752131477427069, 4.39341200215651142863346213800, 6.02766128913803040087682365233, 6.87227119804449168402433114928, 7.45229391530804132732506630172, 8.778682186987611101862945437419, 9.569635716340567359025428112705, 11.33107206972996331373815753498, 11.53486047182417635489845831153
