| L(s) = 1 | + (−0.743 + 0.669i)2-s + (−1.53 − 0.802i)3-s + (0.104 − 0.994i)4-s + (1.99 + 1.01i)5-s + (1.67 − 0.430i)6-s + (1.01 + 0.586i)7-s + (0.587 + 0.809i)8-s + (1.71 + 2.46i)9-s + (−2.15 + 0.582i)10-s + (−4.11 − 4.57i)11-s + (−0.958 + 1.44i)12-s + (2.61 + 2.35i)13-s + (−1.14 + 0.243i)14-s + (−2.24 − 3.15i)15-s + (−0.978 − 0.207i)16-s + (0.704 + 0.969i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 + 0.473i)2-s + (−0.886 − 0.463i)3-s + (0.0522 − 0.497i)4-s + (0.891 + 0.452i)5-s + (0.684 − 0.175i)6-s + (0.384 + 0.221i)7-s + (0.207 + 0.286i)8-s + (0.570 + 0.821i)9-s + (−0.682 + 0.184i)10-s + (−1.24 − 1.37i)11-s + (−0.276 + 0.416i)12-s + (0.724 + 0.652i)13-s + (−0.306 + 0.0651i)14-s + (−0.580 − 0.814i)15-s + (−0.244 − 0.0519i)16-s + (0.170 + 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945674 + 0.230398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945674 + 0.230398i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (1.53 + 0.802i)T \) |
| 5 | \( 1 + (-1.99 - 1.01i)T \) |
| good | 7 | \( 1 + (-1.01 - 0.586i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.11 + 4.57i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.61 - 2.35i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.704 - 0.969i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.98 + 1.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.953 - 4.48i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-5.63 + 2.50i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-5.04 - 2.24i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-9.29 + 3.02i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.09 - 2.33i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-8.90 - 5.14i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.54 - 3.48i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.85 - 3.93i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.620 - 0.688i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.950 + 1.05i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.932 + 2.09i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (2.15 + 1.56i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (16.1 + 5.26i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.65 + 3.85i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (6.47 - 0.680i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.798 + 2.45i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.58 - 10.3i)T + (-64.9 + 72.0i)T^{2} \) |
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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
−11.02444636323630104697358692554, −10.42556434844550943764230789841, −9.381077118596338211227605998658, −8.275404671852379889174526635876, −7.48047063453545715287489530319, −6.22680290462970058299741737798, −5.89304921321161132372315146672, −4.86757701314850690163053281283, −2.73102946549937247496182775334, −1.19526065479008500299612384235,
1.05008015634875236173965134936, 2.61922749611596246770120479736, 4.41319719133115710196915987565, 5.15750175572486753620891092187, 6.21828140510474664314519600353, 7.43158544360864238966182007059, 8.459498139093815676484333166458, 9.595583102523725181412866050157, 10.25957270172866300955640475763, 10.66174440367644719764328365246
