| L(s) = 1 | + (−66.2 + 48.1i)2-s + (155. + 478. i)3-s + (1.44e3 − 4.43e3i)4-s + (−3.90e3 − 2.83e3i)5-s + (−3.33e4 − 2.42e4i)6-s + (1.50e4 − 4.62e4i)7-s + (6.61e4 + 2.03e5i)8-s + (−6.13e4 + 4.45e4i)9-s + 3.95e5·10-s + (3.46e5 − 4.06e5i)11-s + 2.34e6·12-s + (−1.50e6 + 1.09e6i)13-s + (1.23e6 + 3.78e6i)14-s + (7.50e5 − 2.31e6i)15-s + (−6.46e6 − 4.69e6i)16-s + (3.49e6 + 2.54e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 + 1.06i)2-s + (0.369 + 1.13i)3-s + (0.703 − 2.16i)4-s + (−0.559 − 0.406i)5-s + (−1.74 − 1.27i)6-s + (0.337 − 1.03i)7-s + (0.714 + 2.19i)8-s + (−0.346 + 0.251i)9-s + 1.25·10-s + (0.649 − 0.760i)11-s + 2.72·12-s + (−1.12 + 0.817i)13-s + (0.611 + 1.88i)14-s + (0.255 − 0.785i)15-s + (−1.54 − 1.12i)16-s + (0.597 + 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.704650 + 0.0748522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.704650 + 0.0748522i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.46e5 + 4.06e5i)T \) |
| good | 2 | \( 1 + (66.2 - 48.1i)T + (632. - 1.94e3i)T^{2} \) |
| 3 | \( 1 + (-155. - 478. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (3.90e3 + 2.83e3i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-1.50e4 + 4.62e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (1.50e6 - 1.09e6i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-3.49e6 - 2.54e6i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (6.47e6 + 1.99e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 - 2.70e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (8.03e6 - 2.47e7i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-1.57e8 + 1.14e8i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (-1.24e8 + 3.83e8i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (5.08e7 + 1.56e8i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 + 3.85e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (5.45e7 + 1.67e8i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-1.09e9 + 7.97e8i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-1.51e9 + 4.65e9i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (4.21e9 + 3.06e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 + 1.38e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-7.50e9 - 5.44e9i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (-2.29e9 + 7.05e9i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (1.98e10 - 1.43e10i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (4.06e10 + 2.95e10i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 + 6.01e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-9.73e9 + 7.07e9i)T + (2.21e21 - 6.80e21i)T^{2} \) |
| 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
−17.22094844034888178941649013731, −16.61113512971867870414073608170, −15.42514982452394948443028715846, −14.36196936966139128512462281614, −11.03216700899478379523229490659, −9.679653171849157895012561884268, −8.588096559507557649537188593187, −7.03531636004189676444372675301, −4.49264170688110919288039673450, −0.60556050955060572833507921812,
1.44209098339427518029624356235, 2.75904521819617487364033352585, 7.33600291434491530718341824247, 8.320164042148019166056058710719, 10.00351749691591630031462178166, 11.89628301365223986615458557031, 12.46101143002811195094159452047, 14.91746792031186917760337788376, 17.09270626735909224250768424218, 18.31541001218926949273855090388
