| L(s) = 1 | + (−7.91 + 5.74i)2-s + (−77.1 − 237. i)3-s + (−128. + 395. i)4-s + (690. + 501. i)5-s + (1.97e3 + 1.43e3i)6-s + (−2.60e3 + 8.02e3i)7-s + (−2.80e3 − 8.63e3i)8-s + (−3.45e4 + 2.50e4i)9-s − 8.34e3·10-s + (4.06e4 + 2.65e4i)11-s + 1.03e5·12-s + (−7.65e4 + 5.56e4i)13-s + (−2.55e4 − 7.85e4i)14-s + (6.58e4 − 2.02e5i)15-s + (−1.00e5 − 7.31e4i)16-s + (−1.33e5 − 9.70e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.349 + 0.254i)2-s + (−0.549 − 1.69i)3-s + (−0.251 + 0.773i)4-s + (0.493 + 0.358i)5-s + (0.622 + 0.452i)6-s + (−0.410 + 1.26i)7-s + (−0.242 − 0.745i)8-s + (−1.75 + 1.27i)9-s − 0.263·10-s + (0.837 + 0.545i)11-s + 1.44·12-s + (−0.743 + 0.540i)13-s + (−0.177 − 0.546i)14-s + (0.335 − 1.03i)15-s + (−0.383 − 0.278i)16-s + (−0.387 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 - 0.942i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.318792 + 0.451586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.318792 + 0.451586i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.06e4 - 2.65e4i)T \) |
| good | 2 | \( 1 + (7.91 - 5.74i)T + (158. - 486. i)T^{2} \) |
| 3 | \( 1 + (77.1 + 237. i)T + (-1.59e4 + 1.15e4i)T^{2} \) |
| 5 | \( 1 + (-690. - 501. i)T + (6.03e5 + 1.85e6i)T^{2} \) |
| 7 | \( 1 + (2.60e3 - 8.02e3i)T + (-3.26e7 - 2.37e7i)T^{2} \) |
| 13 | \( 1 + (7.65e4 - 5.56e4i)T + (3.27e9 - 1.00e10i)T^{2} \) |
| 17 | \( 1 + (1.33e5 + 9.70e4i)T + (3.66e10 + 1.12e11i)T^{2} \) |
| 19 | \( 1 + (-1.15e5 - 3.54e5i)T + (-2.61e11 + 1.89e11i)T^{2} \) |
| 23 | \( 1 + 1.52e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (1.22e6 - 3.78e6i)T + (-1.17e13 - 8.52e12i)T^{2} \) |
| 31 | \( 1 + (4.54e6 - 3.30e6i)T + (8.17e12 - 2.51e13i)T^{2} \) |
| 37 | \( 1 + (-2.72e6 + 8.38e6i)T + (-1.05e14 - 7.63e13i)T^{2} \) |
| 41 | \( 1 + (-7.69e5 - 2.36e6i)T + (-2.64e14 + 1.92e14i)T^{2} \) |
| 43 | \( 1 - 3.18e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-6.40e6 - 1.97e7i)T + (-9.05e14 + 6.57e14i)T^{2} \) |
| 53 | \( 1 + (-4.19e7 + 3.04e7i)T + (1.01e15 - 3.13e15i)T^{2} \) |
| 59 | \( 1 + (2.87e7 - 8.84e7i)T + (-7.00e15 - 5.09e15i)T^{2} \) |
| 61 | \( 1 + (5.44e7 + 3.95e7i)T + (3.61e15 + 1.11e16i)T^{2} \) |
| 67 | \( 1 + 2.07e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-1.18e8 - 8.59e7i)T + (1.41e16 + 4.36e16i)T^{2} \) |
| 73 | \( 1 + (-1.19e8 + 3.66e8i)T + (-4.76e16 - 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-5.04e7 + 3.66e7i)T + (3.70e16 - 1.13e17i)T^{2} \) |
| 83 | \( 1 + (1.78e7 + 1.29e7i)T + (5.77e16 + 1.77e17i)T^{2} \) |
| 89 | \( 1 + 2.98e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + (9.34e8 - 6.79e8i)T + (2.34e17 - 7.23e17i)T^{2} \) |
| 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
−18.27944923840538049565374220736, −17.82833082703398571671252748244, −16.47326522774636804994252222453, −14.16274572369319190768952682932, −12.54786273297637578656772444290, −12.00759262034620378144864197682, −9.110675470604029803527041505556, −7.36685367879344377355347098384, −6.15832309069958803988511215369, −2.19278634437231328388346660560,
0.38160461203645415406171708605, 4.18072477324358975670150417314, 5.76712802610073233140861845166, 9.309970149951348813652883480586, 10.10444642520720096807112305597, 11.20424121904499616177938570098, 13.84942507199909421795401146297, 15.22265493463247392850682497668, 16.73544063026257516060023058640, 17.44847695634126322919073164227
