| L(s) = 1 | + (−416. + 135. i)2-s + (−4.35e3 − 3.16e3i)3-s + (1.02e5 − 7.43e4i)4-s + (1.61e5 − 4.96e5i)5-s + (2.24e6 + 7.28e5i)6-s + (1.02e6 + 1.40e6i)7-s + (−1.56e7 + 2.15e7i)8-s + (−4.34e6 − 1.33e7i)9-s + 2.28e8i·10-s + (5.44e7 + 2.07e8i)11-s − 6.80e8·12-s + (9.24e8 − 3.00e8i)13-s + (−6.16e8 − 4.47e8i)14-s + (−2.27e9 + 1.65e9i)15-s + (1.05e9 − 3.23e9i)16-s + (2.98e9 + 9.68e8i)17-s + ⋯ |
| L(s) = 1 | + (−1.62 + 0.528i)2-s + (−0.663 − 0.482i)3-s + (1.56 − 1.13i)4-s + (0.413 − 1.27i)5-s + (1.33 + 0.433i)6-s + (0.177 + 0.244i)7-s + (−0.934 + 1.28i)8-s + (−0.101 − 0.310i)9-s + 2.28i·10-s + (0.254 + 0.967i)11-s − 1.58·12-s + (1.13 − 0.368i)13-s + (−0.417 − 0.303i)14-s + (−0.887 + 0.644i)15-s + (0.244 − 0.753i)16-s + (0.427 + 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.492828 - 0.510541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.492828 - 0.510541i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-5.44e7 - 2.07e8i)T \) |
| good | 2 | \( 1 + (416. - 135. i)T + (5.30e4 - 3.85e4i)T^{2} \) |
| 3 | \( 1 + (4.35e3 + 3.16e3i)T + (1.33e7 + 4.09e7i)T^{2} \) |
| 5 | \( 1 + (-1.61e5 + 4.96e5i)T + (-1.23e11 - 8.96e10i)T^{2} \) |
| 7 | \( 1 + (-1.02e6 - 1.40e6i)T + (-1.02e13 + 3.16e13i)T^{2} \) |
| 13 | \( 1 + (-9.24e8 + 3.00e8i)T + (5.38e17 - 3.91e17i)T^{2} \) |
| 17 | \( 1 + (-2.98e9 - 9.68e8i)T + (3.93e19 + 2.86e19i)T^{2} \) |
| 19 | \( 1 + (-1.59e10 + 2.19e10i)T + (-8.91e19 - 2.74e20i)T^{2} \) |
| 23 | \( 1 - 7.00e10T + 6.13e21T^{2} \) |
| 29 | \( 1 + (2.51e11 + 3.46e11i)T + (-7.73e22 + 2.37e23i)T^{2} \) |
| 31 | \( 1 + (-2.78e11 - 8.56e11i)T + (-5.88e23 + 4.27e23i)T^{2} \) |
| 37 | \( 1 + (-2.81e12 + 2.04e12i)T + (3.81e24 - 1.17e25i)T^{2} \) |
| 41 | \( 1 + (-1.41e12 + 1.94e12i)T + (-1.97e25 - 6.06e25i)T^{2} \) |
| 43 | \( 1 + 4.76e12iT - 1.36e26T^{2} \) |
| 47 | \( 1 + (3.05e13 + 2.22e13i)T + (1.75e26 + 5.39e26i)T^{2} \) |
| 53 | \( 1 + (2.57e13 + 7.91e13i)T + (-3.13e27 + 2.27e27i)T^{2} \) |
| 59 | \( 1 + (2.03e14 - 1.48e14i)T + (6.66e27 - 2.05e28i)T^{2} \) |
| 61 | \( 1 + (-1.46e14 - 4.76e13i)T + (2.97e28 + 2.16e28i)T^{2} \) |
| 67 | \( 1 - 4.47e14T + 1.64e29T^{2} \) |
| 71 | \( 1 + (1.69e13 - 5.21e13i)T + (-3.37e29 - 2.45e29i)T^{2} \) |
| 73 | \( 1 + (-3.13e14 - 4.31e14i)T + (-2.00e29 + 6.18e29i)T^{2} \) |
| 79 | \( 1 + (2.16e15 - 7.04e14i)T + (1.86e30 - 1.35e30i)T^{2} \) |
| 83 | \( 1 + (-2.08e15 - 6.77e14i)T + (4.10e30 + 2.98e30i)T^{2} \) |
| 89 | \( 1 + 7.65e15T + 1.54e31T^{2} \) |
| 97 | \( 1 + (2.77e15 + 8.54e15i)T + (-4.96e31 + 3.61e31i)T^{2} \) |
| 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
−16.65430149206200509865397224923, −15.38772184045635096801555488599, −12.91692788739358360526995044780, −11.41172861890336189287724555364, −9.576370222508023513030493230498, −8.588830954182160019813299777195, −6.93261608773926872503114122283, −5.44841947673094557408402091395, −1.43558455378931034907639137164, −0.62344332779631175177197775899,
1.25316816746007144309063653525, 3.11408394492692183664874920262, 6.15746763309802803996139756015, 7.894448026530530107551288734013, 9.661058496500370544639768952893, 10.87647214921957426078708084701, 11.31820423395405501759163971126, 14.08292187282039488405206494563, 16.16233127571674686409589507919, 17.02419907416971957544476808418
