| L(s) = 1 | + (−0.115 + 0.429i)2-s + (−13.5 − 7.79i)3-s + (13.6 + 7.90i)4-s + (3.39 + 12.6i)5-s + (4.90 − 4.90i)6-s + (−35.4 + 61.4i)7-s + (−9.99 + 9.99i)8-s + (81.0 + 140. i)9-s − 5.82·10-s + 139. i·11-s + (−123. − 213. i)12-s + (−51.0 − 190. i)13-s + (−22.2 − 22.2i)14-s + (52.8 − 197. i)15-s + (123. + 213. i)16-s + (−95.3 − 25.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0287 + 0.107i)2-s + (−1.50 − 0.866i)3-s + (0.855 + 0.493i)4-s + (0.135 + 0.506i)5-s + (0.136 − 0.136i)6-s + (−0.723 + 1.25i)7-s + (−0.156 + 0.156i)8-s + (1.00 + 1.73i)9-s − 0.0582·10-s + 1.15i·11-s + (−0.855 − 1.48i)12-s + (−0.302 − 1.12i)13-s + (−0.113 − 0.113i)14-s + (0.234 − 0.876i)15-s + (0.481 + 0.834i)16-s + (−0.330 − 0.0884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.487668 + 0.579018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487668 + 0.579018i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-841. - 1.07e3i)T \) |
| good | 2 | \( 1 + (0.115 - 0.429i)T + (-13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (13.5 + 7.79i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-3.39 - 12.6i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (35.4 - 61.4i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 139. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (51.0 + 190. i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (95.3 + 25.5i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (9.54 + 35.6i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (637. - 637. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (762. + 762. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-517. - 517. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-967. - 558. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-707. + 707. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.01e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (647. + 1.12e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (2.00e3 + 537. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-3.79e3 + 1.01e3i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-7.24e3 - 4.18e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (793. - 1.37e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 1.30e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.90e3 + 1.08e4i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (869. + 1.50e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.00e3 - 1.12e4i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (647. - 647. i)T - 8.85e7iT^{2} \) |
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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
−15.97683585676299471787728603253, −15.23428872365538832669672930216, −12.92561541706281484695798456306, −12.27229627126511748522537818997, −11.43770742425570602600681530962, −10.06772225186968650024103225187, −7.67841320516023562586570254626, −6.56229538680984496767723803297, −5.64004210379857904271414736119, −2.35897962135177205308562283653,
0.60335774442388945224632082196, 4.16933786634300427415010720676, 5.83413500876849643072711734722, 6.81955128392029269886534965717, 9.514663643413211047542340579729, 10.61074646551999444955608485948, 11.25399168384053779020971180643, 12.53798775435361594209976051410, 14.21783488398656574456938406837, 15.90379743134762951166765704666
