| L(s) = 1 | + (−5.37 + 6.20i)2-s + (−13.1 + 8.42i)3-s + (−0.485 − 3.37i)4-s + (−45.2 + 20.6i)5-s + (18.2 − 126. i)6-s + (−13.8 − 47.0i)7-s + (−418. − 268. i)8-s + (100. − 221. i)9-s + (115. − 391. i)10-s + (464. − 402. i)11-s + (34.8 + 40.1i)12-s + (−2.25e3 − 661. i)13-s + (365. + 167. i)14-s + (419. − 652. i)15-s + (4.12e3 − 1.21e3i)16-s + (−6.77e3 − 974. i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.775i)2-s + (−0.485 + 0.312i)3-s + (−0.00758 − 0.0527i)4-s + (−0.361 + 0.165i)5-s + (0.0843 − 0.586i)6-s + (−0.0402 − 0.137i)7-s + (−0.817 − 0.525i)8-s + (0.138 − 0.303i)9-s + (0.115 − 0.391i)10-s + (0.348 − 0.302i)11-s + (0.0201 + 0.0232i)12-s + (−1.02 − 0.300i)13-s + (0.133 + 0.0609i)14-s + (0.124 − 0.193i)15-s + (1.00 − 0.295i)16-s + (−1.37 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.634980 + 0.0445869i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.634980 + 0.0445869i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.1 - 8.42i)T \) |
| 23 | \( 1 + (-1.21e4 + 272. i)T \) |
| good | 2 | \( 1 + (5.37 - 6.20i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (45.2 - 20.6i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (13.8 + 47.0i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-464. + 402. i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (2.25e3 + 661. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (6.77e3 + 974. i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-1.20e4 + 1.73e3i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (954. - 6.63e3i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-3.93e4 - 2.52e4i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-6.31e4 - 2.88e4i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (4.02e4 + 8.81e4i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (5.64e4 + 8.77e4i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 + 5.46e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (5.35e3 + 1.82e4i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (2.88e5 + 8.46e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-1.77e5 + 2.76e5i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (5.16e4 + 4.47e4i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-6.92e4 + 7.99e4i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (6.02e4 + 4.18e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (2.87e4 - 9.78e4i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-3.87e5 - 1.76e5i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (-4.09e5 - 6.37e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-4.89e5 + 2.23e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−13.64383262706100907901509094642, −12.20507393224466065106721689023, −11.32269857463802535285549806812, −9.863705071683369485267715931034, −8.876705655102114606074652798893, −7.49028498364530934975730366999, −6.65617192221603895925290388967, −5.04305655626843733507504726493, −3.26033692612128877972948580115, −0.42771382870305009906196857517,
0.996254556462644057948574084628, 2.53398062742400634025049433724, 4.71132400222778990669275538990, 6.28020165486092354692252349093, 7.73970496422076334058380440408, 9.208704814097028192480668152422, 10.06683138445031506768487842379, 11.51484659415350422464664122661, 11.82312324928215287415077090898, 13.18627450106795478093869990095
