| L(s) = 1 | + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.654 − 0.755i)6-s + (0.455 − 3.16i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.142 + 0.989i)10-s + (−1.70 − 3.74i)11-s + (−0.415 − 0.909i)12-s + (0.401 + 2.79i)13-s + (−2.69 − 1.72i)14-s + (−0.959 + 0.281i)15-s + (−0.142 + 0.989i)16-s + (5.33 − 6.15i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.376 + 0.241i)5-s + (0.267 − 0.308i)6-s + (0.172 − 1.19i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.0450 + 0.313i)10-s + (−0.515 − 1.12i)11-s + (−0.119 − 0.262i)12-s + (0.111 + 0.774i)13-s + (−0.719 − 0.462i)14-s + (−0.247 + 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (1.29 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.990142 - 1.45286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990142 - 1.45286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.79 - 0.138i)T \) |
| good | 7 | \( 1 + (-0.455 + 3.16i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.70 + 3.74i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.401 - 2.79i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-5.33 + 6.15i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.54 + 1.78i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-2.60 + 3.00i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (1.90 - 0.558i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.51 - 0.970i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-2.69 + 1.73i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-5.02 - 1.47i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 + (-0.655 + 4.55i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.277 + 1.92i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-10.9 + 3.21i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (2.01 - 4.40i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.33 - 9.48i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-8.35 - 9.63i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.92 - 13.3i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (3.43 + 2.20i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (10.2 + 3.00i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (10.0 - 6.47i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.19951757082769197262097657642, −9.618035835811879990560917691124, −8.451938416549158402602243916230, −7.70352036434726436169371850158, −6.78332225240443220091481683097, −5.44932134833782619879349772887, −4.32199629315187596088941089113, −3.57262881499687628580145265555, −2.57467066943881102907481858182, −0.821371160515337860463081977098,
1.93131047533423196594365687610, 3.23527348863993454173250606917, 4.34228307965916640506462297435, 5.46078578036087336937972203420, 6.13792924450717741655266997160, 7.51692757113286276938240360384, 8.057009848949325381925431266160, 8.684960000537602319890793837740, 9.745381935606385467936001320084, 10.54116531774222743411882280120
