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 | \( 1 - 1.52e5iT - 8.32e11T^{2} \) |
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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
−10.48006490680997584816752493346, −9.250368787573007983795727351418, −8.593206366208987074913068401727, −7.72606399892841875477138967648, −6.46870009858502424812845912665, −5.65185284917344568792411093285, −4.39348514337082035444664554925, −2.85141723512364117705107652468, −2.20002014452831837892392668107, −0.65729820613407984486957917116,
1.08511488486799431367275422982, 2.21653202491271264589236806145, 3.63840994992874162014850783797, 4.57367942028304611824732441009, 5.62467292374443269461153005130, 7.09990798800779904755509928816, 7.72421134355124174249961320110, 8.913997189408460567446893800276, 9.707334678190781709753334542100, 10.54421962504838632836143623001