L(s) = 1 | + (13.5 + 7.79i)3-s + (44.9 − 25.9i)5-s + (120. − 321. i)7-s + (121.5 + 210. i)9-s + (−453. + 785. i)11-s + 510. i·13-s + 809.·15-s + (−566. − 327. i)17-s + (−2.88e3 + 1.66e3i)19-s + (4.13e3 − 3.39e3i)21-s + (9.70e3 + 1.68e4i)23-s + (−6.46e3 + 1.11e4i)25-s + 3.78e3i·27-s + 6.03e3·29-s + (−8.30e3 − 4.79e3i)31-s + ⋯ |
L(s) = 1 | + (0.5 + 0.288i)3-s + (0.359 − 0.207i)5-s + (0.352 − 0.935i)7-s + (0.166 + 0.288i)9-s + (−0.340 + 0.590i)11-s + 0.232i·13-s + 0.239·15-s + (−0.115 − 0.0666i)17-s + (−0.420 + 0.242i)19-s + (0.446 − 0.366i)21-s + (0.797 + 1.38i)23-s + (−0.413 + 0.716i)25-s + 0.192i·27-s + 0.247·29-s + (−0.278 − 0.160i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.740117633\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.740117633\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 - 7.79i)T \) |
| 7 | \( 1 + (-120. + 321. i)T \) |
good | 5 | \( 1 + (-44.9 + 25.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (453. - 785. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 510. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (566. + 327. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (2.88e3 - 1.66e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-9.70e3 - 1.68e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 6.03e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (8.30e3 + 4.79e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.89e4 - 5.01e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 4.30e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.25e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.76e5 + 1.01e5i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (4.26e4 - 7.37e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.08e5 - 1.20e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.85e5 - 1.06e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.46e5 + 2.53e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 6.22e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (9.70e4 + 5.60e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.77e5 - 6.54e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 9.10e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (5.01e5 - 2.89e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.52e5iT - 8.32e11T^{2} \) |
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
−10.54421962504838632836143623001, −9.707334678190781709753334542100, −8.913997189408460567446893800276, −7.72421134355124174249961320110, −7.09990798800779904755509928816, −5.62467292374443269461153005130, −4.57367942028304611824732441009, −3.63840994992874162014850783797, −2.21653202491271264589236806145, −1.08511488486799431367275422982,
0.65729820613407984486957917116, 2.20002014452831837892392668107, 2.85141723512364117705107652468, 4.39348514337082035444664554925, 5.65185284917344568792411093285, 6.46870009858502424812845912665, 7.72606399892841875477138967648, 8.593206366208987074913068401727, 9.250368787573007983795727351418, 10.48006490680997584816752493346