L(s) = 1 | + (1.73 + 3i)2-s + (3.5 + 6.06i)3-s + (−2 + 3.46i)4-s − 13.8·5-s + (−12.1 + 21i)6-s + (−11.2 + 19.5i)7-s + 13.8·8-s + (−11 + 19.0i)9-s + (−23.9 − 41.5i)10-s + (−11.2 − 19.5i)11-s − 28.0·12-s − 78·14-s + (−48.4 − 84i)15-s + (39.9 + 69.2i)16-s + (13.5 − 23.3i)17-s − 76.2·18-s + ⋯ |
L(s) = 1 | + (0.612 + 1.06i)2-s + (0.673 + 1.16i)3-s + (−0.250 + 0.433i)4-s − 1.23·5-s + (−0.824 + 1.42i)6-s + (−0.607 + 1.05i)7-s + 0.612·8-s + (−0.407 + 0.705i)9-s + (−0.758 − 1.31i)10-s + (−0.308 − 0.534i)11-s − 0.673·12-s − 1.48·14-s + (−0.834 − 1.44i)15-s + (0.624 + 1.08i)16-s + (0.192 − 0.333i)17-s − 0.997·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.335163 - 1.99670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.335163 - 1.99670i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-1.73 - 3i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 + (11.2 - 19.5i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (11.2 + 19.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-13.5 + 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (44.1 - 76.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-28.5 - 49.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-34.5 - 59.7i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + (19.9 + 34.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-196. - 340.5i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (42.5 - 73.6i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-9.52 + 16.5i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-8.5 + 14.7i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (82.2 + 142.5i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-291. + 505.5i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-153. - 265.5i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-617. + 1.06e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.08077271646546253250017404290, −11.98962217258746655753109634959, −10.80894277250198753617968164107, −9.677857687529950795393269751824, −8.546305185147663937398107420501, −7.81685575874874361855510550476, −6.38352627037604110932431769694, −5.21215179297313945104997192734, −4.10360630340555716848383515633, −3.12004335684016734339341771507,
0.71893077149200097815950516107, 2.36986999567970864789516699977, 3.55471853931236530034756333824, 4.50593432421718688574708695636, 6.89820921126313904293861866205, 7.47224532272548667089043245728, 8.418041462657932276202687146163, 10.12276725304448429124908136488, 11.04556752892983942830455106656, 12.06569434718210944740294938151