| L(s) = 1 | + (−4.25 − 1.54i)2-s + (−41.1 + 14.9i)3-s + (−82.3 − 69.1i)4-s + (−92.7 + 525. i)5-s + 197.·6-s + (9.66 − 54.8i)7-s + (532. + 922. i)8-s + (−209. + 175. i)9-s + (1.20e3 − 2.09e3i)10-s + (−2.79e3 − 4.84e3i)11-s + (4.42e3 + 1.60e3i)12-s + (−1.37e3 − 1.15e3i)13-s + (−125. + 218. i)14-s + (−4.05e3 − 2.30e4i)15-s + (1.55e3 + 8.81e3i)16-s + (1.16e4 − 9.75e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.375 − 0.136i)2-s + (−0.878 + 0.319i)3-s + (−0.643 − 0.540i)4-s + (−0.331 + 1.88i)5-s + 0.373·6-s + (0.0106 − 0.0604i)7-s + (0.367 + 0.637i)8-s + (−0.0958 + 0.0804i)9-s + (0.381 − 0.661i)10-s + (−0.633 − 1.09i)11-s + (0.738 + 0.268i)12-s + (−0.173 − 0.145i)13-s + (−0.0122 + 0.0212i)14-s + (−0.310 − 1.76i)15-s + (0.0948 + 0.537i)16-s + (0.574 − 0.481i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.256351 - 0.189307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256351 - 0.189307i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (3.32e4 - 3.06e5i)T \) |
| good | 2 | \( 1 + (4.25 + 1.54i)T + (98.0 + 82.2i)T^{2} \) |
| 3 | \( 1 + (41.1 - 14.9i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (92.7 - 525. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-9.66 + 54.8i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (2.79e3 + 4.84e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.37e3 + 1.15e3i)T + (1.08e7 + 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-1.16e4 + 9.75e3i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 19 | \( 1 + (-4.46e3 + 1.62e3i)T + (6.84e8 - 5.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.05e4 + 3.55e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-9.51e4 - 1.64e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 9.36e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-4.09e4 - 3.43e4i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + 8.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.71e5 + 2.96e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.78e5 + 1.00e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (4.44e5 + 2.51e6i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (2.91e5 + 2.44e5i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.90e5 + 1.64e6i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-3.23e5 + 1.17e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 - 3.99e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-4.37e5 + 2.48e6i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (4.43e6 - 3.72e6i)T + (4.71e12 - 2.67e13i)T^{2} \) |
| 89 | \( 1 + (2.20e6 + 1.25e7i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-1.94e5 + 3.36e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−14.54565082062935314757659900295, −13.76003173334994553552173648498, −11.58563259797546007009463920492, −10.71984932705472272594810413422, −10.15631431803789640539287126226, −8.166023825106859456505902152190, −6.48672221290529061385728407974, −5.16926087874564501001273208298, −3.03413932947797708426114042577, −0.23935885280213290660558459748,
0.998079742461217317391455221560, 4.35270255005629158709027161440, 5.43215751709942509029765528457, 7.56711855135058435568811423345, 8.668890669490585626529694487840, 9.819427472381388933957449841358, 11.97025199988515550186107826708, 12.45893429319409861136734160096, 13.41785975167900561456141555943, 15.52334560785796632566129215929
