| L(s) = 1 | + (−8.00 + 4.62i)2-s + (−13.9 + 24.0i)3-s + (−21.3 + 36.9i)4-s + (−360. − 208. i)5-s − 257. i·6-s + (67.3 − 116. i)7-s − 1.57e3i·8-s + (706. + 1.22e3i)9-s + 3.84e3·10-s − 8.59e3·11-s + (−592. − 1.02e3i)12-s + (7.67e3 + 4.42e3i)13-s + 1.24e3i·14-s + (1.00e4 − 5.79e3i)15-s + (4.55e3 + 7.89e3i)16-s + (1.78e4 − 1.02e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.408i)2-s + (−0.297 + 0.515i)3-s + (−0.166 + 0.288i)4-s + (−1.28 − 0.744i)5-s − 0.485i·6-s + (0.0742 − 0.128i)7-s − 1.08i·8-s + (0.323 + 0.559i)9-s + 1.21·10-s − 1.94·11-s + (−0.0990 − 0.171i)12-s + (0.968 + 0.559i)13-s + 0.121i·14-s + (0.767 − 0.443i)15-s + (0.278 + 0.481i)16-s + (0.880 − 0.508i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.522021 - 0.0259574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522021 - 0.0259574i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (3.99e4 + 3.05e5i)T \) |
| good | 2 | \( 1 + (8.00 - 4.62i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (13.9 - 24.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (360. + 208. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-67.3 + 116. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 8.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-7.67e3 - 4.42e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.78e4 + 1.02e4i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.59e4 - 2.07e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + 3.21e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 4.93e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.84e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (2.13e5 - 3.69e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + 2.92e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.73e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (6.43e5 + 1.11e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-2.36e6 + 1.36e6i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.09e6 + 1.20e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.21e5 + 1.59e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.03e6 + 1.80e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 5.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (8.65e5 + 4.99e5i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.20e6 - 3.81e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.13e6 + 6.55e5i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 7.70e6iT - 8.07e13T^{2} \) |
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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
−15.63462998383582703830953820889, −13.47456322470424601059561223351, −12.37445844098775648471894648579, −10.99793066684070583486005848170, −9.674638450288525095259200401555, −8.083134412333148119909324817346, −7.66332056744816286939639881048, −5.11182966622699986999963880146, −3.77542133547003176554538166710, −0.44768022843674832392461576927,
0.904408031636220698629981328222, 3.16614113892684915361737855987, 5.46096651781933747658464560928, 7.36782352756039289808731249569, 8.325521100566702426137376769128, 10.16519598160269986885362231670, 11.04962522339622877949625437255, 12.12436717098548380301172518302, 13.55767706660745402127717747654, 15.18714044115682109906515546934
