| 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 | \( 1 + (-10.0 + 1.44i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.46443446365055123134588662567, −9.680733000485467398397464954506, −8.596453035070181269548306408999, −8.423017906351144251226195652662, −7.32528504195601318604026589614, −6.19225231118571252283168235560, −4.97129678628683726887321461960, −3.96466198861896076833991665364, −2.31416321617305235578625976934, −0.76727090317425903006067311370,
1.64819534026317787750513155762, 2.78762452035583997152332153267, 3.76797094752990529502128089116, 5.91520383869514800712177137740, 6.56389541015121674609371438218, 7.53293303367374072749720916835, 8.333621940189062134158193415322, 8.988681299496498104898537629832, 10.15248869234962673929617118581, 10.80385001810113224903588465220
