| L(s) = 1 | + (0.889 + 0.699i)2-s + (−0.814 − 0.580i)3-s + (−0.169 − 0.698i)4-s + (1.17 − 0.225i)5-s + (−0.318 − 1.08i)6-s + (−2.56 − 0.637i)7-s + (1.27 − 2.79i)8-s + (0.327 + 0.945i)9-s + (1.20 + 0.618i)10-s + (−1.31 − 1.66i)11-s + (−0.267 + 0.666i)12-s + (−0.833 − 1.29i)13-s + (−1.83 − 2.36i)14-s + (−1.08 − 0.495i)15-s + (1.81 − 0.937i)16-s + (−1.55 − 1.48i)17-s + ⋯ |
| L(s) = 1 | + (0.629 + 0.494i)2-s + (−0.470 − 0.334i)3-s + (−0.0846 − 0.349i)4-s + (0.523 − 0.100i)5-s + (−0.130 − 0.443i)6-s + (−0.970 − 0.240i)7-s + (0.451 − 0.989i)8-s + (0.109 + 0.315i)9-s + (0.379 + 0.195i)10-s + (−0.395 − 0.503i)11-s + (−0.0770 + 0.192i)12-s + (−0.231 − 0.359i)13-s + (−0.491 − 0.631i)14-s + (−0.280 − 0.127i)15-s + (0.454 − 0.234i)16-s + (−0.376 − 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01073 - 0.910445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01073 - 0.910445i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 + 0.580i)T \) |
| 7 | \( 1 + (2.56 + 0.637i)T \) |
| 23 | \( 1 + (1.40 + 4.58i)T \) |
| good | 2 | \( 1 + (-0.889 - 0.699i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.17 + 0.225i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (1.31 + 1.66i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.833 + 1.29i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.55 + 1.48i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 1.59i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-8.02 + 2.35i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.0120 - 0.125i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-7.68 + 2.66i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (8.08 - 7.00i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.11 + 2.79i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-2.18 - 1.26i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.23 - 0.154i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (2.28 - 4.42i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 9.15i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (3.51 + 8.78i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (0.570 - 3.96i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.102 - 0.0247i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-2.00 + 0.0952i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.929 + 1.07i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.806 + 0.0770i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-10.9 - 12.6i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.57921420468702151392256734916, −10.04228045819079002423120839009, −9.134558346887693031101132083353, −7.75168660226447875139290797547, −6.69432391492842985645654647372, −6.11620746339574523491793836637, −5.29811414061530272949460651772, −4.24917808026170114576780390924, −2.70537160129580846678266656388, −0.70888213281362090668373920097,
2.18305717351189558175784051451, 3.33482109054327216566249537269, 4.37672617436306339626969542285, 5.40900546249617865449972533503, 6.32759489486515090995358582037, 7.44078264328519470987676440581, 8.644385753319778307703992349368, 9.727556371227608115188490909802, 10.26756804936962297171443393129, 11.38641863682016297250423011562
