L(s) = 1 | + (1.41 − 0.0470i)2-s + (0.881 − 0.471i)3-s + (1.99 − 0.132i)4-s + (−2.41 − 2.94i)5-s + (1.22 − 0.707i)6-s + (−0.253 + 0.169i)7-s + (2.81 − 0.281i)8-s + (0.555 − 0.831i)9-s + (−3.55 − 4.04i)10-s + (−0.265 + 0.0805i)11-s + (1.69 − 1.05i)12-s + (1.32 + 1.08i)13-s + (−0.350 + 0.251i)14-s + (−3.52 − 1.45i)15-s + (3.96 − 0.530i)16-s + (4.25 − 1.76i)17-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0332i)2-s + (0.509 − 0.272i)3-s + (0.997 − 0.0664i)4-s + (−1.08 − 1.31i)5-s + (0.499 − 0.288i)6-s + (−0.0958 + 0.0640i)7-s + (0.995 − 0.0996i)8-s + (0.185 − 0.277i)9-s + (−1.12 − 1.28i)10-s + (−0.0800 + 0.0242i)11-s + (0.489 − 0.305i)12-s + (0.366 + 0.300i)13-s + (−0.0936 + 0.0671i)14-s + (−0.908 − 0.376i)15-s + (0.991 − 0.132i)16-s + (1.03 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20624 - 1.19712i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20624 - 1.19712i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0470i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
good | 5 | \( 1 + (2.41 + 2.94i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (0.253 - 0.169i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.265 - 0.0805i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 1.08i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.25 + 1.76i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.116 - 1.17i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (7.69 - 1.52i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.259 + 0.854i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-7.54 - 7.54i)T + 31iT^{2} \) |
| 37 | \( 1 + (11.1 + 1.10i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.635 - 3.19i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-6.03 - 3.22i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-3.17 - 7.67i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.522 + 1.72i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-1.79 + 1.47i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.97 + 7.44i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.29 + 2.41i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.04 + 6.05i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (3.57 + 2.38i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.38 + 5.75i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-9.70 + 0.955i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (6.03 + 1.20i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-7.57 - 7.57i)T + 97iT^{2} \) |
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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
−11.78158563518892914938961776635, −10.43491578501360449531387633947, −9.227541400835066777121189924307, −8.097997627400999438301904858373, −7.63972131377295712512071490867, −6.28312891655544750549127865717, −5.08871864097803301187313216425, −4.17254443465159067776751226258, −3.22636255438627287010467720271, −1.41613520679695779163136000148,
2.48485383589087831771952561546, 3.53654100886983553186899873336, 4.11834641351778206750447739750, 5.69289409129356547319840108097, 6.75459060177965203526720472606, 7.63497036445739824382269804381, 8.313920726212771880038790460723, 10.20846602060616128683058979632, 10.52987584388842317803822080427, 11.69894080524760524536313677990