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 | \( 1 + (-1.08e5 + 7.87e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.91302809011375762882592332773, −17.77031815464258247633934634369, −16.39850517838538335826718582864, −15.46063543348171096048024396910, −12.85732854987115256156920710611, −11.78192233575174191278302716776, −8.963976807570103129485698927172, −7.87714962784561615865364335275, −6.24879423684032138894150288349, −0.35262732083598023687836627605,
3.91137272689274126697257756700, 7.64592495914858090483041005725, 10.01100456787731917301877102256, 10.47574952505201901448061094339, 11.88887381011350717624838039602, 14.71094343531759850434564794099, 16.15765046817417820432463062719, 17.32945423960902489542752558722, 18.88133451372181988914635868445, 19.97160449697029364981980689209