| L(s) = 1 | + (−13.5 − 7.79i)3-s + (71.9 − 41.5i)5-s + (−77.0 − 334. i)7-s + (121.5 + 210. i)9-s + (−221. + 383. i)11-s + 696. i·13-s − 1.29e3·15-s + (−5.44e3 − 3.14e3i)17-s + (2.32e3 − 1.34e3i)19-s + (−1.56e3 + 5.11e3i)21-s + (7.88e3 + 1.36e4i)23-s + (−4.36e3 + 7.55e3i)25-s − 3.78e3i·27-s − 2.32e4·29-s + (−4.11e4 − 2.37e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (0.575 − 0.332i)5-s + (−0.224 − 0.974i)7-s + (0.166 + 0.288i)9-s + (−0.166 + 0.287i)11-s + 0.317i·13-s − 0.383·15-s + (−1.10 − 0.640i)17-s + (0.339 − 0.195i)19-s + (−0.168 + 0.552i)21-s + (0.648 + 1.12i)23-s + (−0.279 + 0.483i)25-s − 0.192i·27-s − 0.954·29-s + (−1.38 − 0.798i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7310018877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7310018877\) |
| \(L(4)\) |
|
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 | \( 1 + (77.0 + 334. i)T \) |
| good | 5 | \( 1 + (-71.9 + 41.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (221. - 383. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 696. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (5.44e3 + 3.14e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-2.32e3 + 1.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.88e3 - 1.36e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.32e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (4.11e4 + 2.37e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-5.07e3 - 8.79e3i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.51e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.35e4 + 2.51e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-9.97e4 + 1.72e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (3.35e5 + 1.93e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.02e4 - 5.89e3i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.92e5 - 3.32e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.25e5 - 1.87e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.65e4 - 2.87e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 9.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (2.27e5 - 1.31e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 5.75e5iT - 8.32e11T^{2} \) |
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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
−10.95691865202522367541675059749, −9.676321199524272277073501911072, −9.179869064485480925859687973864, −7.56386139915876485123873861994, −7.03437993233992608240111138417, −5.84988004454782717232627309415, −4.91083564410473213952709539287, −3.77010745740463946006115566441, −2.14583541093227216815973048636, −1.01325477623767329999587408429,
0.19852958630913159070343652399, 1.91508317858871652958182522996, 2.98751327908550103171531506217, 4.40108428999162731502547868452, 5.67147386542925040397339359564, 6.13013011362382187372899869982, 7.33090862876998424684930612781, 8.753766600376589247065288893285, 9.308397883628473499867207539224, 10.57765032695737174255179869759
