L(s) = 1 | + (−6.36 − 4.62i)2-s + (−5.98 + 18.4i)3-s + (9.23 + 28.4i)4-s + (−59.3 + 43.1i)5-s + (123. − 89.5i)6-s + (−51.6 − 158. i)7-s + (−5.17 + 15.9i)8-s + (−106. − 77.6i)9-s + 576.·10-s + (3.42 + 401. i)11-s − 578.·12-s + (−139. − 101. i)13-s + (−405. + 1.24e3i)14-s + (−438. − 1.35e3i)15-s + (879. − 639. i)16-s + (365. − 265. i)17-s + ⋯ |
L(s) = 1 | + (−1.12 − 0.817i)2-s + (−0.383 + 1.18i)3-s + (0.288 + 0.887i)4-s + (−1.06 + 0.771i)5-s + (1.39 − 1.01i)6-s + (−0.398 − 1.22i)7-s + (−0.0285 + 0.0879i)8-s + (−0.439 − 0.319i)9-s + 1.82·10-s + (0.00853 + 0.999i)11-s − 1.15·12-s + (−0.228 − 0.166i)13-s + (−0.553 + 1.70i)14-s + (−0.503 − 1.55i)15-s + (0.859 − 0.624i)16-s + (0.306 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.108914 + 0.222573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.108914 + 0.222573i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-3.42 - 401. i)T \) |
good | 2 | \( 1 + (6.36 + 4.62i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (5.98 - 18.4i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + (59.3 - 43.1i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (51.6 + 158. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (139. + 101. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-365. + 265. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (764. - 2.35e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (709. + 2.18e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-3.78e3 - 2.74e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-951. - 2.92e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.21e3 + 6.81e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 398.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-415. + 1.27e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (8.16e3 + 5.93e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.22e4 - 3.77e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.36e4 + 9.89e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.91e4 - 3.57e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.10e4 - 3.39e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.01e4 - 7.35e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.22e4 + 8.91e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.08e5 - 7.87e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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
−19.97160449697029364981980689209, −18.88133451372181988914635868445, −17.32945423960902489542752558722, −16.15765046817417820432463062719, −14.71094343531759850434564794099, −11.88887381011350717624838039602, −10.47574952505201901448061094339, −10.01100456787731917301877102256, −7.64592495914858090483041005725, −3.91137272689274126697257756700,
0.35262732083598023687836627605, 6.24879423684032138894150288349, 7.87714962784561615865364335275, 8.963976807570103129485698927172, 11.78192233575174191278302716776, 12.85732854987115256156920710611, 15.46063543348171096048024396910, 16.39850517838538335826718582864, 17.77031815464258247633934634369, 18.91302809011375762882592332773