| L(s) = 1 | + (−7.46 + 8.61i)2-s + (13.1 − 8.42i)3-s + (−9.39 − 65.3i)4-s + (−12.8 + 5.88i)5-s + (−25.2 + 175. i)6-s + (−31.2 − 106. i)7-s + (19.5 + 12.5i)8-s + (100. − 221. i)9-s + (45.4 − 154. i)10-s + (−461. + 399. i)11-s + (−674. − 778. i)12-s + (2.95e3 + 868. i)13-s + (1.15e3 + 525. i)14-s + (−119. + 185. i)15-s + (3.80e3 − 1.11e3i)16-s + (2.16e3 + 310. i)17-s + ⋯ |
| L(s) = 1 | + (−0.933 + 1.07i)2-s + (0.485 − 0.312i)3-s + (−0.146 − 1.02i)4-s + (−0.103 + 0.0470i)5-s + (−0.117 + 0.814i)6-s + (−0.0910 − 0.310i)7-s + (0.0382 + 0.0245i)8-s + (0.138 − 0.303i)9-s + (0.0454 − 0.154i)10-s + (−0.346 + 0.300i)11-s + (−0.390 − 0.450i)12-s + (1.34 + 0.395i)13-s + (0.419 + 0.191i)14-s + (−0.0353 + 0.0550i)15-s + (0.927 − 0.272i)16-s + (0.439 + 0.0632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.799604 + 0.904829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799604 + 0.904829i\) |
| \(L(4)\) |
|
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 | \( 1 + (-67.1 - 1.21e4i)T \) |
| good | 2 | \( 1 + (7.46 - 8.61i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (12.8 - 5.88i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (31.2 + 106. i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (461. - 399. i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-2.95e3 - 868. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-2.16e3 - 310. i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-5.27e3 + 758. i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (3.01e3 - 2.09e4i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (6.35e3 + 4.08e3i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-3.54e4 - 1.61e4i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-3.16e4 - 6.92e4i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (1.34e4 + 2.09e4i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 - 1.73e3T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-3.98e4 - 1.35e5i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.65e5 - 4.85e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-1.01e5 + 1.57e5i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (-2.16e5 - 1.87e5i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (5.49e3 - 6.33e3i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (2.80e4 + 1.94e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (9.89e4 - 3.36e5i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (7.12e3 + 3.25e3i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (3.10e5 + 4.82e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-2.43e5 + 1.11e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−14.01519842807994558672729022601, −12.98839441277598596298698420672, −11.45126905176852677664407764341, −9.920286267708890216057514664262, −8.955974979576317017317748367839, −7.86600684111884051696579182062, −7.05290226150766043483792527963, −5.74475815696814476347437016575, −3.51551438195658121199041190766, −1.18135170601835095076385169953,
0.74795478929788104052229848034, 2.42547689834384769676405839927, 3.69353353432702885783003107705, 5.80758631221364443817634154838, 7.969483409020201265354328050268, 8.772193620178779461175749601855, 9.871725283567464570531721717729, 10.77394459761333700943428248691, 11.78997168986412254475170717962, 12.97744531200369460564286377535
