| L(s) = 1 | + (1.21 − 3.33i)2-s + (−1.70 + 0.300i)3-s + (−6.57 − 5.51i)4-s + (−0.783 + 0.657i)5-s + (−1.06 + 6.05i)6-s + (4.99 − 8.64i)7-s + (−14.0 + 8.13i)8-s + (2.81 − 1.02i)9-s + (1.24 + 3.40i)10-s + (7.98 + 13.8i)11-s + (12.8 + 7.43i)12-s + (16.0 + 2.83i)13-s + (−22.7 − 27.1i)14-s + (1.13 − 1.35i)15-s + (4.05 + 23.0i)16-s + (−10.7 − 3.90i)17-s + ⋯ |
| L(s) = 1 | + (0.606 − 1.66i)2-s + (−0.568 + 0.100i)3-s + (−1.64 − 1.37i)4-s + (−0.156 + 0.131i)5-s + (−0.177 + 1.00i)6-s + (0.712 − 1.23i)7-s + (−1.76 + 1.01i)8-s + (0.313 − 0.114i)9-s + (0.124 + 0.340i)10-s + (0.725 + 1.25i)11-s + (1.07 + 0.619i)12-s + (1.23 + 0.217i)13-s + (−1.62 − 1.93i)14-s + (0.0758 − 0.0904i)15-s + (0.253 + 1.43i)16-s + (−0.630 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.424380 - 1.20728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424380 - 1.20728i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 - 0.300i)T \) |
| 19 | \( 1 + (13.5 + 13.2i)T \) |
| good | 2 | \( 1 + (-1.21 + 3.33i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (0.783 - 0.657i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.99 + 8.64i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.98 - 13.8i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-16.0 - 2.83i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (10.7 + 3.90i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-3.44 - 2.88i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-12.7 - 35.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-1.66 - 0.962i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 23.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (57.7 - 10.1i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-61.4 + 51.5i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-41.7 + 15.1i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (39.0 - 46.5i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-9.63 + 26.4i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (29.6 + 24.9i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-1.49 - 4.09i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (13.1 + 15.6i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (7.70 + 43.7i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (11.1 - 1.97i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (12.2 - 21.2i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-26.3 - 4.65i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (51.6 - 141. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.06765833656746399358727108258, −13.22368520352245833509150258440, −12.05702596558808373538027865827, −11.05347743519261960092640060128, −10.55449187845222313708971230129, −9.138873950658192391605131833712, −6.93727281610509658735503725016, −4.78619202249019796090117456299, −3.90148625660842839381610314787, −1.44209496214367552976017817212,
4.21553840212248018897279318711, 5.78554993656693283742650460003, 6.28565803141039666368613728424, 8.198810331948050147802837883028, 8.742846689112931576314356217397, 11.15279844493795272410505073739, 12.29440267987671436781258380688, 13.51004302499663801541500952688, 14.50508489543521625042138762312, 15.56014991476887933156271016561
