L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.987 − 1.13i)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 + (1.26 − 0.815i)10-s + (−0.650 − 4.52i)11-s + (−0.142 − 0.989i)12-s + (0.448 − 0.288i)13-s + (−0.654 + 0.755i)14-s + (0.626 − 1.37i)15-s + (0.841 + 0.540i)16-s + (1.56 − 0.458i)17-s + ⋯ |
L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.441 − 0.509i)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 + (0.401 − 0.257i)10-s + (−0.196 − 1.36i)11-s + (−0.0410 − 0.285i)12-s + (0.124 − 0.0799i)13-s + (−0.175 + 0.201i)14-s + (0.161 − 0.354i)15-s + (0.210 + 0.135i)16-s + (0.379 − 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34388 + 0.0243134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34388 + 0.0243134i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.679 + 4.74i)T \) |
good | 5 | \( 1 + (0.987 + 1.13i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.650 + 4.52i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.448 + 0.288i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 0.458i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.620 - 0.182i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-6.91 + 2.03i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.297 + 0.650i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.04 - 1.20i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.692 + 0.799i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.60 - 3.51i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 9.54T + 47T^{2} \) |
| 53 | \( 1 + (2.74 + 1.76i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-9.57 + 6.15i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.50 + 3.30i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.277 - 1.92i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0910 - 0.633i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.14 + 1.21i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.05 - 1.96i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (2.89 - 3.34i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.97 + 8.71i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.30 - 7.27i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.939655197799830915088536346835, −8.831073759655041097488269871492, −8.413563664857030566441066710426, −7.81697496192618591525958852752, −6.52934858126714000943082037131, −5.67078121042325560859266023218, −4.80870931664717617807549069145, −3.94407293125754195436739425830, −2.77289181875434238446097939018, −0.71398228728253780410710518368,
1.33104633817796507027003102641, 2.46205126984138766085148916418, 3.52518735406867521906746177335, 4.50002275594961700976725916123, 5.59207790726000735208567944805, 7.01327577370468717564647073982, 7.42855517192099226381388076857, 8.332703123438285043685982397704, 9.276556201063185174832470466682, 10.13201253603577674278215379955