| L(s) = 1 | + (12.6 + 21.8i)2-s + (−378. − 654. i)3-s + (705. − 1.22e3i)4-s − 3.12e3·5-s + (9.53e3 − 1.65e4i)6-s + (−3.42e4 + 5.93e4i)7-s + 8.72e4·8-s + (−1.97e5 + 3.41e5i)9-s + (−3.93e4 − 6.81e4i)10-s + (−5.63e4 − 9.76e4i)11-s − 1.06e6·12-s + (3.93e5 − 1.27e6i)13-s − 1.72e6·14-s + (1.17e6 + 2.04e6i)15-s + (−3.44e5 − 5.97e5i)16-s + (−5.20e6 + 9.01e6i)17-s + ⋯ |
| L(s) = 1 | + (0.278 + 0.482i)2-s + (−0.898 − 1.55i)3-s + (0.344 − 0.596i)4-s − 0.446·5-s + (0.500 − 0.867i)6-s + (−0.770 + 1.33i)7-s + 0.941·8-s + (−1.11 + 1.92i)9-s + (−0.124 − 0.215i)10-s + (−0.105 − 0.182i)11-s − 1.23·12-s + (0.294 − 0.955i)13-s − 0.859·14-s + (0.401 + 0.694i)15-s + (−0.0821 − 0.142i)16-s + (−0.888 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0362606 + 0.127533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0362606 + 0.127533i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-3.93e5 + 1.27e6i)T \) |
| good | 2 | \( 1 + (-12.6 - 21.8i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (378. + 654. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + 3.12e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + (3.42e4 - 5.93e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (5.63e4 + 9.76e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 17 | \( 1 + (5.20e6 - 9.01e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.21e6 - 3.83e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.49e7 + 2.59e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (1.58e7 + 2.73e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + 9.05e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (1.60e8 + 2.78e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-1.22e8 - 2.12e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (7.30e7 - 1.26e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 1.13e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.45e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-3.20e9 + 5.54e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.27e9 - 9.13e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.24e9 + 2.15e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + (-1.26e10 + 2.19e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 + 2.58e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.15e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.66e9 - 4.60e9i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (5.18e10 - 8.98e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
| 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
−16.26364276953589302802584142164, −15.08740673155954692645492964563, −13.19055269882266805648414391451, −12.25782912609911883198257108892, −10.83692906346691867188837335328, −8.076685997535519511421481537330, −6.38875272647485416113907699166, −5.74255764595556134771739277624, −1.98616203234363753809692791654, −0.05993679524666296611042266000,
3.57765411067225466072090984025, 4.49378669052265908573672267541, 6.95186331304650679913981181945, 9.594656812267213185928835192418, 10.92017427637869798184256285489, 11.73442376883942498389906539684, 13.61007755901994271681791202218, 15.80038438821027915407232982958, 16.34481973192069000263137673935, 17.39847825138904358165727450927
