| L(s) = 1 | + (−18.1 + 10.4i)2-s + (−200. − 347. i)3-s + (−805. + 1.39e3i)4-s + 5.33e3i·5-s + (7.26e3 + 4.19e3i)6-s + (−1.50e4 − 8.69e3i)7-s − 7.64e4i·8-s + (8.11e3 − 1.40e4i)9-s + (−5.58e4 − 9.66e4i)10-s + (3.43e5 − 1.98e5i)11-s + 6.46e5·12-s + (1.33e6 + 9.20e4i)13-s + 3.63e5·14-s + (1.85e6 − 1.07e6i)15-s + (−8.50e5 − 1.47e6i)16-s + (3.31e6 − 5.73e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.399 + 0.230i)2-s + (−0.476 − 0.825i)3-s + (−0.393 + 0.681i)4-s + 0.764i·5-s + (0.381 + 0.220i)6-s + (−0.338 − 0.195i)7-s − 0.825i·8-s + (0.0458 − 0.0793i)9-s + (−0.176 − 0.305i)10-s + (0.642 − 0.371i)11-s + 0.749·12-s + (0.997 + 0.0687i)13-s + 0.180·14-s + (0.630 − 0.364i)15-s + (−0.202 − 0.351i)16-s + (0.565 − 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.965938 - 0.301587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.965938 - 0.301587i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.33e6 - 9.20e4i)T \) |
| good | 2 | \( 1 + (18.1 - 10.4i)T + (1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (200. + 347. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 5.33e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 + (1.50e4 + 8.69e3i)T + (9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-3.43e5 + 1.98e5i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (-3.31e6 + 5.73e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-7.69e5 - 4.44e5i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-9.87e6 - 1.70e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (3.14e7 + 5.45e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 - 1.91e7iT - 2.54e16T^{2} \) |
| 37 | \( 1 + (-1.13e8 + 6.55e7i)T + (8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-1.22e9 + 7.09e8i)T + (2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-8.04e7 + 1.39e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 - 1.75e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 1.14e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (4.20e9 + 2.42e9i)T + (1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-5.21e9 + 9.02e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.19e10 + 6.90e9i)T + (6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (3.26e9 + 1.88e9i)T + (1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 + 2.37e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 3.62e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.76e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.86e8 + 1.07e8i)T + (1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (7.46e9 + 4.31e9i)T + (3.57e21 + 6.19e21i)T^{2} \) |
| 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
−17.28172603643986017132963507813, −15.98984305080672897644159285383, −13.98523619966947170766127665701, −12.72328738762168221011006241588, −11.35644492801990094959825130009, −9.344175675852362036828206654305, −7.51215989360393755477998143328, −6.39884636659539106403976074664, −3.49348616772364483300288047852, −0.74322969366189839774964710505,
1.20158903511858046273626197072, 4.36909730364087935665201100445, 5.79439195037239274472127337702, 8.698162415274312826974494069269, 9.887222702422354874623304955115, 11.08770185832520047930142757067, 12.92487566704890602912507950118, 14.69963824772562986681793317143, 16.10730055175388685891263725380, 17.16266435492910575460319117570
