| L(s) = 1 | + (9.14 − 5.28i)2-s + (2.08 − 3.60i)3-s + (−8.24 + 14.2i)4-s + (−276. − 159. i)5-s − 43.9i·6-s + (−330. + 572. i)7-s + 1.52e3i·8-s + (1.08e3 + 1.87e3i)9-s − 3.37e3·10-s − 3.39e3·11-s + (34.3 + 59.4i)12-s + (−3.26e3 − 1.88e3i)13-s + 6.98e3i·14-s + (−1.15e3 + 664. i)15-s + (7.00e3 + 1.21e4i)16-s + (−1.45e4 + 8.37e3i)17-s + ⋯ |
| L(s) = 1 | + (0.808 − 0.466i)2-s + (0.0445 − 0.0771i)3-s + (−0.0643 + 0.111i)4-s + (−0.989 − 0.571i)5-s − 0.0831i·6-s + (−0.364 + 0.631i)7-s + 1.05i·8-s + (0.496 + 0.859i)9-s − 1.06·10-s − 0.768·11-s + (0.00573 + 0.00993i)12-s + (−0.412 − 0.237i)13-s + 0.680i·14-s + (−0.0880 + 0.0508i)15-s + (0.427 + 0.740i)16-s + (−0.716 + 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.496495 + 0.775631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496495 + 0.775631i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (2.43e5 - 1.89e5i)T \) |
| good | 2 | \( 1 + (-9.14 + 5.28i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-2.08 + 3.60i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (276. + 159. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (330. - 572. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 3.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.26e3 + 1.88e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.45e4 - 8.37e3i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.80e3 + 1.04e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 1.46e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 4.58e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 3.11e4iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (-3.60e5 + 6.23e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + 2.95e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-5.52e3 - 9.56e3i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (2.00e6 - 1.16e6i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.79e6 - 1.03e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.57e6 - 2.72e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.24e6 - 2.16e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 9.24e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-6.87e6 - 3.96e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (5.60e3 + 9.70e3i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-6.31e6 + 3.64e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.03e7iT - 8.07e13T^{2} \) |
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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
−15.28353026608540321886530582944, −13.65016720156748119982894818429, −12.73426244923149444975042539169, −11.99783624750001719651631559243, −10.65549830789983531877983857503, −8.687177285660079232442876023080, −7.64040281702174391852267331308, −5.29024302641810507621972811357, −4.13002811324418820858715606011, −2.45387147599720243635077965887,
0.30481420671700887110198744118, 3.45986912908059738624109270569, 4.61096484484101491433995076043, 6.51479070223091117259646803005, 7.46459335616287714942436498553, 9.529194802322464150163339093998, 10.80062764380710567636028611052, 12.33573759118921791748628157254, 13.41892161920880297613083131674, 14.65402032794625268185710230713
