| L(s) = 1 | + (−0.877 + 0.479i)2-s + (−0.997 + 0.0713i)3-s + (0.540 − 0.841i)4-s + (0.565 − 2.16i)5-s + (0.841 − 0.540i)6-s + (1.25 − 3.36i)7-s + (−0.0713 + 0.997i)8-s + (0.989 − 0.142i)9-s + (0.540 + 2.16i)10-s + (0.643 + 2.19i)11-s + (−0.479 + 0.877i)12-s + (2.56 − 0.954i)13-s + (0.511 + 3.55i)14-s + (−0.410 + 2.19i)15-s + (−0.415 − 0.909i)16-s + (0.483 + 0.105i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 + 0.338i)2-s + (−0.575 + 0.0411i)3-s + (0.270 − 0.420i)4-s + (0.253 − 0.967i)5-s + (0.343 − 0.220i)6-s + (0.474 − 1.27i)7-s + (−0.0252 + 0.352i)8-s + (0.329 − 0.0474i)9-s + (0.170 + 0.686i)10-s + (0.194 + 0.660i)11-s + (−0.138 + 0.253i)12-s + (0.710 − 0.264i)13-s + (0.136 + 0.950i)14-s + (−0.105 + 0.567i)15-s + (−0.103 − 0.227i)16-s + (0.117 + 0.0255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0187 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0187 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620048 - 0.631766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620048 - 0.631766i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.877 - 0.479i)T \) |
| 3 | \( 1 + (0.997 - 0.0713i)T \) |
| 5 | \( 1 + (-0.565 + 2.16i)T \) |
| 23 | \( 1 + (3.80 + 2.92i)T \) |
| good | 7 | \( 1 + (-1.25 + 3.36i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (-0.643 - 2.19i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.56 + 0.954i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (-0.483 - 0.105i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (2.66 + 1.71i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 2.19i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.17 + 2.51i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-2.70 + 3.60i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (1.05 - 7.32i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.0953 + 1.33i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (-6.96 + 6.96i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.48 + 1.29i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 0.799i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.99 + 6.06i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.79 + 6.95i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (8.39 + 2.46i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (2.60 + 11.9i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (3.20 - 7.00i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (2.35 + 1.76i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (10.3 - 11.8i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-12.4 + 9.34i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.27062059754958478547687399390, −9.428378941796630101521974634219, −8.460089525847497490518518278180, −7.70466606376439832437607935656, −6.77195204700831857546115955307, −5.85752193434652208530277662317, −4.76846743134783980610211627438, −4.05106071530946155977035771578, −1.78668618279241028937564692486, −0.64211846914451208911592003932,
1.66510933777898173593584064023, 2.77700862658722291377089448275, 4.02206724820083696835304837390, 5.69008480903960964362877047753, 6.10173806940208840850873184170, 7.19893341053159785899948084186, 8.268493543744170761538551713940, 8.962012073899262849588354461209, 9.973647833454336010989636708224, 10.73069303938425386019557543587
