| L(s) = 1 | + (−9.34 + 16.1i)2-s + (−43.2 − 74.8i)3-s + (−110. − 191. i)4-s + (−171. − 297. i)5-s + 1.61e3·6-s + (−666. − 1.15e3i)7-s + 1.74e3·8-s + (−2.64e3 + 4.58e3i)9-s + 6.42e3·10-s − 421.·11-s + (−9.56e3 + 1.65e4i)12-s + (−3.73e3 − 6.46e3i)13-s + 2.48e4·14-s + (−1.48e4 + 2.57e4i)15-s + (−2.10e3 + 3.64e3i)16-s + (1.08e4 − 1.87e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.825 + 1.43i)2-s + (−0.924 − 1.60i)3-s + (−0.863 − 1.49i)4-s + (−0.614 − 1.06i)5-s + 3.05·6-s + (−0.734 − 1.27i)7-s + 1.20·8-s + (−1.20 + 2.09i)9-s + 2.03·10-s − 0.0954·11-s + (−1.59 + 2.76i)12-s + (−0.471 − 0.816i)13-s + 2.42·14-s + (−1.13 + 1.96i)15-s + (−0.128 + 0.222i)16-s + (0.535 − 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0898732 + 0.246334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0898732 + 0.246334i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-9.78e4 + 2.92e5i)T \) |
| good | 2 | \( 1 + (9.34 - 16.1i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (43.2 + 74.8i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (171. + 297. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (666. + 1.15e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + 421.T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.73e3 + 6.46e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.08e4 + 1.87e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.76e3 + 4.78e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + 8.33e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.83e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.15e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-1.08e5 - 1.87e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + 6.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-9.99e5 + 1.73e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.32e6 + 2.29e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.39e5 + 5.88e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-5.40e5 - 9.36e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.23e6 + 3.87e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 9.77e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.71e6 - 2.96e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.23e5 + 7.33e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.44e6 - 2.49e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 4.54e6T + 8.07e13T^{2} \) |
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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
−14.05294548445784838676332231988, −12.97416116158148358460133527912, −11.96049944588525716866595994732, −10.09680925102051105484951334358, −8.193178995122471243281337704470, −7.49456726043940395321546721481, −6.52172867953226028980457228480, −5.16239183843332195489360665539, −0.71324420387829282438842909366, −0.29144418754973021219807739824,
2.83492924093026295168941620392, 3.99768291957662701965522865453, 6.08233328222793096451819455704, 8.737263016800774914948253973580, 9.924771961530119108542072258790, 10.47623026703743416410807282706, 11.76546144712938653536606408600, 12.08490397012149850579620366452, 14.80982477221813222519891812681, 15.70000067012999294940909144036
