| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (2.42 + 0.712i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (−3.14 + 3.63i)11-s + (0.654 − 0.755i)12-s + (3.41 − 1.00i)13-s + (−1.05 − 2.30i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.684 + 4.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (0.0580 + 0.404i)6-s + (0.917 + 0.269i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (−0.948 + 1.09i)11-s + (0.189 − 0.218i)12-s + (0.948 − 0.278i)13-s + (−0.280 − 0.614i)14-s + (−0.217 + 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.166 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08062 - 0.0515694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08062 - 0.0515694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.83 - 4.43i)T \) |
| good | 7 | \( 1 + (-2.42 - 0.712i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (3.14 - 3.63i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.41 + 1.00i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.684 - 4.76i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.0293 - 0.203i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.860 - 5.98i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.39 + 3.46i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 3.50i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.17 + 9.13i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.62 - 1.04i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + (-12.5 - 3.67i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-7.41 + 2.17i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.22 + 5.92i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.71 - 7.74i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (1.47 + 1.69i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.534 + 3.71i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (9.22 - 2.70i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.629 - 1.37i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (11.8 + 7.63i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.356 + 0.780i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.46749063425684776845784591924, −9.826663843818898463395701123220, −8.554544703641615657927416779596, −8.087892793324294878209090732145, −7.14396547957273450047803379512, −5.83235474941149723040631737202, −5.04087586805542807946406828737, −3.93342611890817486872665828045, −2.26186658698314954171373848639, −1.30326820172393733088022027656,
0.830308835255917802358902774400, 2.66253314915598495893033912218, 4.20916593461424285185141933670, 5.23701413368049013942146273013, 6.02798261329302801838813162700, 6.91767198626224189369234935633, 8.035114496075917973658969802123, 8.525596052665584503209787412130, 9.733523836125661639151274542594, 10.47259089068469381834800718498
