| L(s) = 1 | + (−1.39 − 0.227i)2-s + (0.654 − 0.755i)3-s + (1.89 + 0.635i)4-s + (−0.355 + 2.46i)5-s + (−1.08 + 0.905i)6-s + (0.661 + 1.44i)7-s + (−2.50 − 1.31i)8-s + (−0.142 − 0.989i)9-s + (1.05 − 3.36i)10-s + (−0.328 − 1.12i)11-s + (1.72 − 1.01i)12-s + (3.21 + 1.47i)13-s + (−0.593 − 2.17i)14-s + (1.63 + 1.88i)15-s + (3.19 + 2.41i)16-s + (−0.248 − 0.387i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 − 0.161i)2-s + (0.378 − 0.436i)3-s + (0.948 + 0.317i)4-s + (−0.158 + 1.10i)5-s + (−0.443 + 0.369i)6-s + (0.250 + 0.547i)7-s + (−0.884 − 0.466i)8-s + (−0.0474 − 0.329i)9-s + (0.334 − 1.06i)10-s + (−0.0991 − 0.337i)11-s + (0.497 − 0.293i)12-s + (0.892 + 0.407i)13-s + (−0.158 − 0.580i)14-s + (0.421 + 0.486i)15-s + (0.797 + 0.602i)16-s + (−0.0603 − 0.0939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.847775 + 0.492056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847775 + 0.492056i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.227i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (4.16 - 2.38i)T \) |
| good | 5 | \( 1 + (0.355 - 2.46i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.661 - 1.44i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.328 + 1.12i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.21 - 1.47i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.248 + 0.387i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.28 - 5.10i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.40 - 5.30i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.449 + 0.389i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0548 + 0.381i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.738 - 5.13i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.16 - 1.87i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 0.152iT - 47T^{2} \) |
| 53 | \( 1 + (-3.00 - 6.59i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.844 + 1.85i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.17 - 7.12i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.430 + 1.46i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.422 + 1.43i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (9.07 + 5.83i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.40 + 7.46i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-4.37 + 0.628i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.04 + 1.76i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 1.44i)T + (93.0 + 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80385001810113224903588465220, −10.15248869234962673929617118581, −8.988681299496498104898537629832, −8.333621940189062134158193415322, −7.53293303367374072749720916835, −6.56389541015121674609371438218, −5.91520383869514800712177137740, −3.76797094752990529502128089116, −2.78762452035583997152332153267, −1.64819534026317787750513155762,
0.76727090317425903006067311370, 2.31416321617305235578625976934, 3.96466198861896076833991665364, 4.97129678628683726887321461960, 6.19225231118571252283168235560, 7.32528504195601318604026589614, 8.423017906351144251226195652662, 8.596453035070181269548306408999, 9.680733000485467398397464954506, 10.46443446365055123134588662567
