L(s) = 1 | + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.150 − 0.173i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.193 + 0.124i)10-s + (−0.117 − 0.820i)11-s + (−0.142 − 0.989i)12-s + (2.13 − 1.37i)13-s + (−0.654 + 0.755i)14-s + (0.0955 − 0.209i)15-s + (0.841 + 0.540i)16-s + (3.16 − 0.928i)17-s + ⋯ |
L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.0673 − 0.0777i)5-s + (0.391 − 0.115i)6-s + (−0.317 − 0.204i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.0611 + 0.0393i)10-s + (−0.0355 − 0.247i)11-s + (−0.0410 − 0.285i)12-s + (0.593 − 0.381i)13-s + (−0.175 + 0.201i)14-s + (0.0246 − 0.0540i)15-s + (0.210 + 0.135i)16-s + (0.767 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36144 - 0.913457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36144 - 0.913457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.22 + 2.27i)T \) |
good | 5 | \( 1 + (0.150 + 0.173i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.117 + 0.820i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.13 + 1.37i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.16 + 0.928i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.67 - 0.492i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.74 - 0.805i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.53 + 5.54i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.36 + 2.72i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.88 + 2.17i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.86 - 6.27i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.09T + 47T^{2} \) |
| 53 | \( 1 + (-1.19 - 0.767i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.59 - 2.30i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.88 + 8.49i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.30 + 9.09i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.768 - 5.34i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.777 - 0.228i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (0.112 - 0.0720i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (2.71 - 3.13i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (2.22 + 4.86i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.78 + 3.21i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.912436323078967870742392910722, −9.235975056973070779272539351728, −8.379745488426142707726354142127, −7.55270671967024583695331876038, −6.24055818654268864957520377288, −5.35036194198942204450151100375, −4.31551160338519487038116435708, −3.45264486435376465406275569442, −2.56508142863981499449929702674, −0.849746953267475281136798311338,
1.30252229582268094167934264366, 2.93482893101183574189185918114, 3.87523347931753573633782385834, 5.18908825308213218934875441398, 5.94870419705976171548111852687, 6.95390550384376357546778594343, 7.44545677192873537714616234730, 8.474776517138949185912411395924, 9.116548495753436147316285960204, 9.949382258524240775173396499041