| L(s) = 1 | + (13.5 + 7.79i)3-s + (−126. + 72.9i)5-s + (−114. + 323. i)7-s + (121.5 + 210. i)9-s + (−1.04e3 + 1.80e3i)11-s + 2.13e3i·13-s − 2.27e3·15-s + (2.55e3 + 1.47e3i)17-s + (−3.67e3 + 2.12e3i)19-s + (−4.06e3 + 3.47e3i)21-s + (1.81e3 + 3.13e3i)23-s + (2.81e3 − 4.87e3i)25-s + 3.78e3i·27-s − 257.·29-s + (4.05e4 + 2.34e4i)31-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.288i)3-s + (−1.01 + 0.583i)5-s + (−0.333 + 0.942i)7-s + (0.166 + 0.288i)9-s + (−0.781 + 1.35i)11-s + 0.971i·13-s − 0.673·15-s + (0.519 + 0.300i)17-s + (−0.536 + 0.309i)19-s + (−0.438 + 0.375i)21-s + (0.148 + 0.257i)23-s + (0.180 − 0.312i)25-s + 0.192i·27-s − 0.0105·29-s + (1.36 + 0.785i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.087049525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087049525\) |
| \(L(4)\) |
|
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 | \( 1 + (114. - 323. i)T \) |
| good | 5 | \( 1 + (126. - 72.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (1.04e3 - 1.80e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 2.13e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.55e3 - 1.47e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (3.67e3 - 2.12e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.81e3 - 3.13e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 257.T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.05e4 - 2.34e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.68e4 + 6.37e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 8.98e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.88e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-6.92e4 + 3.99e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (3.29e4 - 5.71e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.30e5 + 7.50e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.73e5 - 1.00e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.28e5 + 3.95e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.65e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (2.36e5 + 1.36e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.06e5 - 5.31e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 4.90e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.17e5 - 3.56e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 2.94e5iT - 8.32e11T^{2} \) |
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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
−11.08424346475785038497603913461, −10.12054880010760531899820121520, −9.289856084356204825128965032396, −8.249936819553272956001857097607, −7.45112326496079221751109417549, −6.48749233548903943937812005395, −5.04140756478464889441140579731, −4.02309005928999514766026456377, −2.94033504559610004811138340472, −1.92367599982504455012876656225,
0.30596546558005855451188157217, 0.849758057344750919421834682865, 2.84615175456530821419101904642, 3.66658472276313731288026419492, 4.78913108669959097028333350383, 6.09648479444752270322513757619, 7.37419027497182936223859514814, 8.086888049686694061584500185668, 8.656975228616813409190429788770, 10.05419399110722735572868951346
