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
−10.57765032695737174255179869759, −9.308397883628473499867207539224, −8.753766600376589247065288893285, −7.33090862876998424684930612781, −6.13013011362382187372899869982, −5.67147386542925040397339359564, −4.40108428999162731502547868452, −2.98751327908550103171531506217, −1.91508317858871652958182522996, −0.19852958630913159070343652399,
1.01325477623767329999587408429, 2.14583541093227216815973048636, 3.77010745740463946006115566441, 4.91083564410473213952709539287, 5.84988004454782717232627309415, 7.03437993233992608240111138417, 7.56386139915876485123873861994, 9.179869064485480925859687973864, 9.676321199524272277073501911072, 10.95691865202522367541675059749