| L(s) = 1 | + (4 − 6.92i)2-s + (−36.1 + 62.6i)3-s + (−31.9 − 55.4i)4-s + (−36.8 + 63.8i)5-s + (289. + 501. i)6-s + 866.·7-s − 511.·8-s + (−1.52e3 − 2.63e3i)9-s + (294. + 510. i)10-s − 8.18e3·11-s + 4.62e3·12-s + (−5.18e3 − 8.98e3i)13-s + (3.46e3 − 6.00e3i)14-s + (−2.66e3 − 4.61e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (1.79e4 − 3.10e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.773 + 1.33i)3-s + (−0.249 − 0.433i)4-s + (−0.131 + 0.228i)5-s + (0.546 + 0.947i)6-s + 0.954·7-s − 0.353·8-s + (−0.695 − 1.20i)9-s + (0.0932 + 0.161i)10-s − 1.85·11-s + 0.773·12-s + (−0.654 − 1.13i)13-s + (0.337 − 0.584i)14-s + (−0.203 − 0.353i)15-s + (−0.125 + 0.216i)16-s + (0.884 − 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0738365 - 0.265349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0738365 - 0.265349i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 19 | \( 1 + (2.82e4 - 9.68e3i)T \) |
| good | 3 | \( 1 + (36.1 - 62.6i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (36.8 - 63.8i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 - 866.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.18e3 + 8.98e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.79e4 + 3.10e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (2.42e4 + 4.20e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.61e3 + 4.52e3i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.03e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.00e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.80e5 - 3.12e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.82e5 - 3.15e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (8.46e4 + 1.46e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (7.38e5 + 1.27e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (7.86e5 - 1.36e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.09e6 + 1.89e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.45e5 - 1.63e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.61e6 - 2.80e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.97e6 - 3.41e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.37e6 + 2.38e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 2.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (9.74e4 + 1.68e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.79e6 - 1.52e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.55739732021886330242353070032, −12.86772170945421136787660448671, −11.51641935678798050571726307602, −10.60628155090362254129667160991, −9.930680169187769462396313494527, −7.945008993901876195999490736873, −5.37403811516712466325679257089, −4.79958254819712539000528722635, −2.91052626785445048864886218792, −0.11289476345704231023888973995,
1.90490011749950398119113337842, 4.81953307305827565827432906071, 6.04377307392773844142324812399, 7.47042627894683546576806771661, 8.248741744549692911043650135947, 10.70016745611218590738908618813, 12.07035772789447939611352419419, 12.80936190606484714863809241058, 13.92973807375466653665339554796, 15.19809555168285033285277345680
