| L(s) = 1 | + (−0.841 − 0.540i)3-s + (−0.102 − 0.713i)5-s + (−0.326 − 0.0311i)7-s + (0.415 + 0.909i)9-s + (4.09 + 1.63i)11-s + (−0.881 − 0.840i)13-s + (−0.299 + 0.655i)15-s + (−0.980 + 2.83i)17-s + (3.20 − 0.306i)19-s + (0.257 + 0.202i)21-s + (0.354 − 7.43i)23-s + (4.29 − 1.26i)25-s + (0.142 − 0.989i)27-s + (1.75 − 3.04i)29-s + (−2.44 + 2.33i)31-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.312i)3-s + (−0.0458 − 0.318i)5-s + (−0.123 − 0.0117i)7-s + (0.138 + 0.303i)9-s + (1.23 + 0.494i)11-s + (−0.244 − 0.232i)13-s + (−0.0772 + 0.169i)15-s + (−0.237 + 0.687i)17-s + (0.736 − 0.0703i)19-s + (0.0562 + 0.0442i)21-s + (0.0738 − 1.55i)23-s + (0.859 − 0.252i)25-s + (0.0273 − 0.190i)27-s + (0.326 − 0.565i)29-s + (−0.439 + 0.418i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23788 - 0.509861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23788 - 0.509861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (-8.16 + 0.538i)T \) |
| good | 5 | \( 1 + (0.102 + 0.713i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.326 + 0.0311i)T + (6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (-4.09 - 1.63i)T + (7.96 + 7.59i)T^{2} \) |
| 13 | \( 1 + (0.881 + 0.840i)T + (0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.980 - 2.83i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (-3.20 + 0.306i)T + (18.6 - 3.59i)T^{2} \) |
| 23 | \( 1 + (-0.354 + 7.43i)T + (-22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.44 - 2.33i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (5.46 + 9.45i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.88 + 1.13i)T + (38.0 - 15.2i)T^{2} \) |
| 43 | \( 1 + (-6.70 - 7.73i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-11.8 + 6.08i)T + (27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (-8.32 + 9.60i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.66 + 0.488i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (10.1 - 4.06i)T + (44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-3.65 - 10.5i)T + (-55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (7.63 - 3.05i)T + (52.8 - 50.3i)T^{2} \) |
| 79 | \( 1 + (-1.42 + 5.88i)T + (-70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (-8.92 + 7.02i)T + (19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (4.70 - 3.02i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.06 + 7.04i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.26245222877486432745697326537, −9.222877603907856056787304434125, −8.569076937128247891631593349825, −7.40593530822919528398248107215, −6.69004496558381939124386428936, −5.83301760965257321567573180820, −4.74446226296271294742050102519, −3.87484710466671039855958797907, −2.31057812105347574987098191171, −0.885713358371290126907543171287,
1.20844813370721724496987823586, 3.00119025728966380418173117678, 3.94244424974424484499811650667, 5.05717243173305870049812644708, 5.97447441620766533157500343488, 6.88298206496527762152244468010, 7.58846846648990461045628790845, 9.109092426014417423726174188144, 9.298551377881409561272969839097, 10.47006608920261977924526948569
