| L(s) = 1 | + (0.750 − 1.68i)2-s + (0.794 − 1.53i)3-s + (−0.942 − 1.04i)4-s + (−2.07 − 0.838i)5-s + (−1.99 − 2.49i)6-s + (−1.99 + 1.15i)7-s + (1.03 − 0.337i)8-s + (−1.73 − 2.44i)9-s + (−2.97 + 2.86i)10-s + (3.59 + 1.59i)11-s + (−2.35 + 0.618i)12-s + (−1.70 − 3.83i)13-s + (0.444 + 4.22i)14-s + (−2.93 + 2.52i)15-s + (0.505 − 4.80i)16-s + (2.60 − 0.844i)17-s + ⋯ |
| L(s) = 1 | + (0.530 − 1.19i)2-s + (0.458 − 0.888i)3-s + (−0.471 − 0.523i)4-s + (−0.927 − 0.374i)5-s + (−0.815 − 1.01i)6-s + (−0.753 + 0.434i)7-s + (0.367 − 0.119i)8-s + (−0.578 − 0.815i)9-s + (−0.939 + 0.906i)10-s + (1.08 + 0.482i)11-s + (−0.681 + 0.178i)12-s + (−0.473 − 1.06i)13-s + (0.118 + 1.12i)14-s + (−0.758 + 0.651i)15-s + (0.126 − 1.20i)16-s + (0.630 − 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.445043 - 1.52551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445043 - 1.52551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.794 + 1.53i)T \) |
| 5 | \( 1 + (2.07 + 0.838i)T \) |
| good | 2 | \( 1 + (-0.750 + 1.68i)T + (-1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (1.99 - 1.15i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.59 - 1.59i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.70 + 3.83i)T + (-8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.60 + 0.844i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 7.29i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-9.40 + 0.988i)T + (22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (1.05 - 0.224i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (6.37 + 1.35i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (4.17 - 5.74i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.49 - 1.55i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.51 + 1.45i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.206 + 0.970i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.16 - 0.379i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.28 - 0.571i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-9.22 - 4.10i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.35 + 6.39i)T + (-61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (0.967 - 2.97i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.376 + 0.517i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.41 - 0.725i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (6.00 + 5.40i)T + (8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (3.52 - 2.55i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.15 - 10.1i)T + (-88.6 + 39.4i)T^{2} \) |
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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
−12.23535159245062365287125402970, −11.35106108364887580860323173243, −10.00377356618810633753779900641, −9.035157809376216542751152782273, −7.79803076607354108783979463533, −6.96873875023770694299390953112, −5.34012379638553806083150385853, −3.69743888764273886830451869345, −3.01873652022894890424789318923, −1.27091804491799343400927779805,
3.29846153111360524147568214375, 4.20023720725464166222772966448, 5.27263992719436359183799231202, 6.87703387970225411757804239657, 7.19856254203509378256562669279, 8.680523159234176549288992120348, 9.428971313263726586452901217859, 10.84077415541501008766518814830, 11.47837986836778722246727153677, 13.00034194920716122762474348530
