| 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
−13.00034194920716122762474348530, −11.47837986836778722246727153677, −10.84077415541501008766518814830, −9.428971313263726586452901217859, −8.680523159234176549288992120348, −7.19856254203509378256562669279, −6.87703387970225411757804239657, −5.27263992719436359183799231202, −4.20023720725464166222772966448, −3.29846153111360524147568214375,
1.27091804491799343400927779805, 3.01873652022894890424789318923, 3.69743888764273886830451869345, 5.34012379638553806083150385853, 6.96873875023770694299390953112, 7.79803076607354108783979463533, 9.035157809376216542751152782273, 10.00377356618810633753779900641, 11.35106108364887580860323173243, 12.23535159245062365287125402970
