| L(s) = 1 | + (−0.686 − 0.727i)2-s + (1.72 − 0.0885i)3-s + (−0.0581 + 0.998i)4-s + (2.21 − 0.258i)5-s + (−1.25 − 1.19i)6-s + (−2.58 − 1.30i)7-s + (0.766 − 0.642i)8-s + (2.98 − 0.306i)9-s + (−1.70 − 1.43i)10-s + (2.24 + 5.19i)11-s + (−0.0121 + 1.73i)12-s + (−3.63 − 0.860i)13-s + (0.830 + 2.77i)14-s + (3.80 − 0.643i)15-s + (−0.993 − 0.116i)16-s + (1.29 − 7.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.514i)2-s + (0.998 − 0.0511i)3-s + (−0.0290 + 0.499i)4-s + (0.989 − 0.115i)5-s + (−0.510 − 0.488i)6-s + (−0.978 − 0.491i)7-s + (0.270 − 0.227i)8-s + (0.994 − 0.102i)9-s + (−0.539 − 0.452i)10-s + (0.675 + 1.56i)11-s + (−0.00351 + 0.499i)12-s + (−1.00 − 0.238i)13-s + (0.222 + 0.741i)14-s + (0.982 − 0.166i)15-s + (−0.248 − 0.0290i)16-s + (0.314 − 1.78i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21309 - 0.407851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21309 - 0.407851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.686 + 0.727i)T \) |
| 3 | \( 1 + (-1.72 + 0.0885i)T \) |
| good | 5 | \( 1 + (-2.21 + 0.258i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (2.58 + 1.30i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-2.24 - 5.19i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (3.63 + 0.860i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-1.29 + 7.36i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.140 + 0.794i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (4.42 - 2.22i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (1.91 - 6.38i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-2.05 - 1.34i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (6.41 - 2.33i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.55 + 2.70i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (6.77 - 9.10i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-3.20 + 2.10i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (1.66 - 2.89i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.15 - 2.67i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.409 + 7.02i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (0.117 + 0.394i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-0.709 - 0.595i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.44 + 2.89i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.27 - 1.34i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (2.38 + 2.53i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-13.5 + 11.3i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.60 - 0.421i)T + (94.3 + 22.3i)T^{2} \) |
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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
−12.75970999482772435047884917872, −12.01374752389441100302801882675, −10.09143470122248724556829583718, −9.753851020936964128122916771141, −9.168481762573844797516360116376, −7.48915955756677231747045614861, −6.84509753965290058436802801260, −4.76827573541535342193368172569, −3.19445685065880598706248438687, −1.90722814677625890108583616024,
2.14775614097493780472675474139, 3.71603956015917593546583110871, 5.84839820626803782236146836631, 6.48962964948932158255571519743, 8.039291630797219150794614974137, 8.899156652253444455135509913446, 9.736836182917140443524877312246, 10.40854271390047671177477663356, 12.16548893044779171735533958380, 13.29657782089538123682566428933
