| L(s) = 1 | + (−4.68 − 1.83i)2-s + (3.18 − 4.10i)3-s + (12.6 + 11.7i)4-s + (5.25 + 3.58i)5-s + (−22.4 + 13.3i)6-s + (−15.9 + 9.47i)7-s + (−20.3 − 42.1i)8-s + (−6.71 − 26.1i)9-s + (−18.0 − 26.4i)10-s + (1.90 + 12.6i)11-s + (88.7 − 14.5i)12-s + (52.6 − 41.9i)13-s + (91.9 − 15.0i)14-s + (31.4 − 10.1i)15-s + (7.24 + 96.6i)16-s + (−94.3 − 29.1i)17-s + ⋯ |
| L(s) = 1 | + (−1.65 − 0.649i)2-s + (0.612 − 0.790i)3-s + (1.58 + 1.47i)4-s + (0.470 + 0.320i)5-s + (−1.52 + 0.909i)6-s + (−0.859 + 0.511i)7-s + (−0.897 − 1.86i)8-s + (−0.248 − 0.968i)9-s + (−0.570 − 0.836i)10-s + (0.0523 + 0.347i)11-s + (2.13 − 0.351i)12-s + (1.12 − 0.895i)13-s + (1.75 − 0.288i)14-s + (0.541 − 0.175i)15-s + (0.113 + 1.51i)16-s + (−1.34 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.256133 - 0.710901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256133 - 0.710901i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.18 + 4.10i)T \) |
| 7 | \( 1 + (15.9 - 9.47i)T \) |
| good | 2 | \( 1 + (4.68 + 1.83i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-5.25 - 3.58i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-1.90 - 12.6i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-52.6 + 41.9i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (94.3 + 29.1i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-126. + 73.0i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (34.7 + 112. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-119. + 27.3i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (142. + 82.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (125. - 116. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-251. + 120. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (156. + 75.4i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (70.9 - 180. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (192. - 207. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-361. + 246. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (183. + 198. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (272. - 471. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-60.6 - 13.8i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-731. + 287. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-62.0 - 107. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-424. + 532. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (303. + 45.7i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.26e3iT - 9.12e5T^{2} \) |
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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
−12.02731061691131637155656411278, −10.98532986289678746967059769524, −9.862228365601749987331021239138, −9.107255424343248709840172714856, −8.325124987149809656458013375876, −7.12664596124369917108221269077, −6.25081665821317125558071003872, −3.12255061651060812610338475819, −2.24782598943644347395036799019, −0.59299789576564783999953033960,
1.57451043048444004524489249595, 3.66225101587616045012961880399, 5.69672878762359527723136701113, 6.84043090512170034229092100789, 8.046711924309378906735484540386, 9.114025115701055007276307988177, 9.472330302963956001809568613067, 10.47925477725324666343479670416, 11.33523975880544300035179089380, 13.36252258077981700688356579346
