L(s) = 1 | + (1 − 1.73i)2-s + (3.16 − 5.48i)3-s + (−1.99 − 3.46i)4-s + (−2.96 + 5.13i)5-s + (−6.33 − 10.9i)6-s + 0.660·7-s − 7.99·8-s + (−6.59 − 11.4i)9-s + (5.93 + 10.2i)10-s + 50.1·11-s − 25.3·12-s + (−1.31 − 2.27i)13-s + (0.660 − 1.14i)14-s + (18.7 + 32.5i)15-s + (−8 + 13.8i)16-s + (−28.9 + 50.0i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.609 − 1.05i)3-s + (−0.249 − 0.433i)4-s + (−0.265 + 0.459i)5-s + (−0.431 − 0.747i)6-s + 0.0356·7-s − 0.353·8-s + (−0.244 − 0.422i)9-s + (0.187 + 0.324i)10-s + 1.37·11-s − 0.609·12-s + (−0.0280 − 0.0485i)13-s + (0.0126 − 0.0218i)14-s + (0.323 + 0.560i)15-s + (−0.125 + 0.216i)16-s + (−0.412 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.23694 - 1.15824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23694 - 1.15824i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 19 | \( 1 + (12.1 - 81.9i)T \) |
good | 3 | \( 1 + (-3.16 + 5.48i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (2.96 - 5.13i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 - 0.660T + 343T^{2} \) |
| 11 | \( 1 - 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + (1.31 + 2.27i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (28.9 - 50.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 23 | \( 1 + (18.9 + 32.8i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (154. + 268. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 21.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-172. + 297. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (132. - 229. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-69.8 - 121. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-66.8 - 115. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (227. - 394. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (54.2 + 93.9i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (13.1 + 22.8i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-559. + 969. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-235. + 407. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (190. - 330. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (202. + 351. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (687. - 1.19e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.96714576204338659308166422905, −14.23224980607800566053637694532, −13.09527876458485700073336870249, −12.11693943263144675395794268589, −10.89241911270170059782718112804, −9.197207684701548141389834757835, −7.71239423243614981784006624429, −6.30573005308637589879292171433, −3.78103340526698265035976608629, −1.83199272898294359011039450711,
3.66518170539807413194052494926, 4.88829527042315552422880682320, 6.91964375884321934312934212643, 8.751524921619016280214616758234, 9.413826969730013822491216420448, 11.26259638405056761568779696269, 12.71137806072286618660885527428, 14.18988913629473380908348363050, 14.92239696507526852869852962833, 16.01168171036747236192244758018