L(s) = 1 | + (0.555 + 0.320i)2-s + (−0.794 − 1.37i)4-s + (1.10 + 1.91i)5-s + (2.60 − 0.458i)7-s − 2.30i·8-s + 1.41i·10-s + (2.93 + 1.69i)11-s + (1.56 − 0.901i)13-s + (1.59 + 0.581i)14-s + (−0.849 + 1.47i)16-s − 5.96·17-s − 1.64i·19-s + (1.75 − 3.04i)20-s + (1.08 + 1.88i)22-s + (−2.05 + 1.18i)23-s + ⋯ |
L(s) = 1 | + (0.392 + 0.226i)2-s + (−0.397 − 0.687i)4-s + (0.494 + 0.856i)5-s + (0.984 − 0.173i)7-s − 0.813i·8-s + 0.448i·10-s + (0.885 + 0.511i)11-s + (0.432 − 0.249i)13-s + (0.426 + 0.155i)14-s + (−0.212 + 0.367i)16-s − 1.44·17-s − 0.377i·19-s + (0.392 − 0.680i)20-s + (0.232 + 0.401i)22-s + (−0.428 + 0.247i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50500 + 0.0761991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50500 + 0.0761991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.60 + 0.458i)T \) |
good | 2 | \( 1 + (-0.555 - 0.320i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 0.901i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 1.64iT - 19T^{2} \) |
| 23 | \( 1 + (2.05 - 1.18i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (9.28 - 5.36i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + (0.455 + 0.788i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.96 - 3.39i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.123 + 0.213i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.87iT - 53T^{2} \) |
| 59 | \( 1 + (5.39 + 9.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.22 + 0.709i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.99 - 6.91i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 0.426iT - 73T^{2} \) |
| 79 | \( 1 + (-2.49 + 4.31i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.28 - 7.42i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.85952223929554062246987584336, −11.35687866283116089849341410560, −10.72488760230915338414251103475, −9.664538595223715725371900061405, −8.686737594234473391528472471615, −7.10238700584501184658462893689, −6.30786315295730248865676933172, −5.08044589968574192994632248007, −3.95678900458276712235893860172, −1.84994846296907642085155337872,
1.90316335246636950398364869551, 3.87575815086337407193920184575, 4.81146582762169884211025334570, 5.97425901353944102708219865840, 7.62676061199001084120187581741, 8.867699523711137581088854198916, 9.039540025726596457443630239269, 10.96005820912406186924493199304, 11.66206705624164096864415814433, 12.62587358770088276914817300987