| L(s) = 1 | + (4 − 6.92i)2-s + (36.4 − 63.2i)3-s + (−31.9 − 55.4i)4-s + (−170. + 295. i)5-s + (−291. − 505. i)6-s − 1.66e3·7-s − 511.·8-s + (−1.56e3 − 2.71e3i)9-s + (1.36e3 + 2.36e3i)10-s + 1.78e3·11-s − 4.67e3·12-s + (−993. − 1.72e3i)13-s + (−6.67e3 + 1.15e4i)14-s + (1.24e4 + 2.15e4i)15-s + (−2.04e3 + 3.54e3i)16-s + (1.31e4 − 2.27e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.780 − 1.35i)3-s + (−0.249 − 0.433i)4-s + (−0.609 + 1.05i)5-s + (−0.551 − 0.955i)6-s − 1.83·7-s − 0.353·8-s + (−0.717 − 1.24i)9-s + (0.431 + 0.746i)10-s + 0.405·11-s − 0.780·12-s + (−0.125 − 0.217i)13-s + (−0.649 + 1.12i)14-s + (0.951 + 1.64i)15-s + (−0.125 + 0.216i)16-s + (0.647 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.281769 + 0.991452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.281769 + 0.991452i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 19 | \( 1 + (2.15e4 + 2.07e4i)T \) |
| good | 3 | \( 1 + (-36.4 + 63.2i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (170. - 295. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 1.66e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (993. + 1.72e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.31e4 + 2.27e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (4.21e4 + 7.29e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-9.36e4 - 1.62e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.82e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.06e5 + 5.31e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.54e5 - 2.68e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (5.05e5 + 8.75e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-3.44e5 - 5.97e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (3.23e5 - 5.60e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.98e5 + 6.89e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.22e6 + 2.11e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.44e6 - 2.50e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (9.03e5 - 1.56e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-3.82e6 + 6.61e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 2.25e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.42e6 - 9.40e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.56e6 + 7.91e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.88332553672540007802355878978, −12.81808267588915147493579879435, −12.08368226629281123481648381530, −10.45278116209385802715143191771, −9.003064498010209590565949163647, −7.21386288222053345385529823838, −6.48550382757736212767416959025, −3.40351744961117569383598240696, −2.59283830430939613623150339755, −0.35287574146464979864726285681,
3.48140071023841265487026098957, 4.23706525274644157466329891067, 6.03190049728154134641621760507, 8.088832289196420985938952921929, 9.212872292035510919052692705924, 10.03269712318890762028014274859, 12.21849821133930190807293275138, 13.21664887580626564515776285933, 14.61717951330343928086167431361, 15.66338042771689069262952143149
