| L(s) = 1 | + (3.18 − 9.79i)2-s + (−234. + 170. i)3-s + (6.54e3 + 4.75e3i)4-s + (−2.05e3 − 6.32e3i)5-s + (921. + 2.83e3i)6-s + (−1.34e5 − 9.74e4i)7-s + (1.35e5 − 9.85e4i)8-s + (−4.66e5 + 1.43e6i)9-s − 6.85e4·10-s + (4.23e6 + 4.07e6i)11-s − 2.33e6·12-s + (−3.60e6 + 1.10e7i)13-s + (−1.38e6 + 1.00e6i)14-s + (1.55e6 + 1.13e6i)15-s + (1.99e7 + 6.13e7i)16-s + (4.56e7 + 1.40e8i)17-s + ⋯ |
| L(s) = 1 | + (0.0351 − 0.108i)2-s + (−0.185 + 0.134i)3-s + (0.798 + 0.580i)4-s + (−0.0588 − 0.181i)5-s + (0.00806 + 0.0248i)6-s + (−0.430 − 0.313i)7-s + (0.182 − 0.132i)8-s + (−0.292 + 0.901i)9-s − 0.0216·10-s + (0.720 + 0.693i)11-s − 0.226·12-s + (−0.206 + 0.637i)13-s + (−0.0490 + 0.0356i)14-s + (0.0352 + 0.0256i)15-s + (0.297 + 0.914i)16-s + (0.458 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0919 - 0.995i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.0919 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.26083 + 1.14972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26083 + 1.14972i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.23e6 - 4.07e6i)T \) |
| good | 2 | \( 1 + (-3.18 + 9.79i)T + (-6.62e3 - 4.81e3i)T^{2} \) |
| 3 | \( 1 + (234. - 170. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (2.05e3 + 6.32e3i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (1.34e5 + 9.74e4i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (3.60e6 - 1.10e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-4.56e7 - 1.40e8i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (1.38e8 - 1.00e8i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 + 3.31e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (2.24e9 + 1.63e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-1.21e9 + 3.73e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-1.64e10 - 1.19e10i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (2.93e10 - 2.13e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 3.09e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-1.65e10 + 1.20e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-7.84e10 + 2.41e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-1.14e11 - 8.33e10i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (3.69e10 + 1.13e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 + 1.22e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-3.95e10 - 1.21e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-1.21e12 - 8.82e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-9.32e11 + 2.86e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (2.56e11 + 7.89e11i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 + 2.66e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (8.31e11 - 2.55e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
| 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
−17.02620236366341566401370911095, −16.52594019897377132676392650659, −14.83500533017701977924337176275, −12.96700917625167887357280005764, −11.68706816149217181215170687912, −10.22343929128471751906381722907, −8.067669853874347700960469570875, −6.43538830114928836559589241033, −4.05776173064674727787271131746, −1.98774893695755445734742302064,
0.76025171009594962558460625390, 2.97557993523451927551797160952, 5.72787776771547211862440661533, 6.97873459596171648191365925876, 9.283854491295147354420420162543, 11.02565789809911418655577060268, 12.25745429199081239489564600844, 14.28852501326546391705775472697, 15.45436794465278659825250744696, 16.75051105476195345904055157469
