| L(s) = 1 | + (0.435 − 0.0207i)2-s + (0.849 − 1.50i)3-s + (−1.80 + 0.172i)4-s + (−0.978 − 0.932i)5-s + (0.338 − 0.674i)6-s + (−0.518 − 2.68i)7-s + (−1.64 + 0.236i)8-s + (−1.55 − 2.56i)9-s + (−0.445 − 0.385i)10-s + (2.26 − 1.16i)11-s + (−1.27 + 2.86i)12-s + (1.92 + 0.370i)13-s + (−0.281 − 1.15i)14-s + (−2.23 + 0.684i)15-s + (2.84 − 0.548i)16-s + (1.01 + 2.22i)17-s + ⋯ |
| L(s) = 1 | + (0.307 − 0.0146i)2-s + (0.490 − 0.871i)3-s + (−0.900 + 0.0860i)4-s + (−0.437 − 0.417i)5-s + (0.138 − 0.275i)6-s + (−0.195 − 1.01i)7-s + (−0.581 + 0.0835i)8-s + (−0.518 − 0.854i)9-s + (−0.140 − 0.122i)10-s + (0.683 − 0.352i)11-s + (−0.366 + 0.827i)12-s + (0.533 + 0.102i)13-s + (−0.0752 − 0.310i)14-s + (−0.578 + 0.176i)15-s + (0.710 − 0.137i)16-s + (0.246 + 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.726905 - 0.912935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726905 - 0.912935i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.849 + 1.50i)T \) |
| 23 | \( 1 + (-1.58 - 4.52i)T \) |
| good | 2 | \( 1 + (-0.435 + 0.0207i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (0.978 + 0.932i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.518 + 2.68i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.16i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 0.370i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.01 - 2.22i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.894 + 0.408i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.476 - 4.98i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-6.18 + 4.86i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (1.37 - 4.67i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.77 - 1.86i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-7.09 + 9.01i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (0.317 + 0.183i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.70 + 3.11i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.79 - 9.33i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-1.63 + 4.07i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-0.0594 + 0.115i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-8.28 - 12.8i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.19 - 4.80i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (13.9 - 4.82i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-5.56 + 5.30i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.57 + 17.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (9.09 + 2.20i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.38172983718893874190170847946, −11.46681682829762980484302449315, −10.01214227515558756736830898037, −8.856573942699986996281378926795, −8.229252649559112549072025969552, −7.09485147041979910877606431422, −5.90478999267700103067260281661, −4.25826264549754385307838563700, −3.43457381159541151746447404959, −0.978413799968160785501014529396,
2.89322148302192744870004476013, 4.00208741866939859078360781933, 5.04643958486155433325889017763, 6.24120531665827170877934517379, 7.960325310134426475459970611417, 8.947245868280971084876270566521, 9.494920265151008268453185931989, 10.63588861107378300522512417275, 11.76775971638041707236075263861, 12.69757781713371407022534649022
