L(s) = 1 | + (−1.37 − 2.37i)2-s + (−1.20 + 2.09i)3-s + (−2.76 + 4.78i)4-s + (−0.312 − 0.540i)5-s + 6.61·6-s + (1.27 + 2.31i)7-s + 9.65·8-s + (−1.41 − 2.44i)9-s + (−0.856 + 1.48i)10-s + (1.23 − 2.13i)11-s + (−6.66 − 11.5i)12-s + (3.76 − 6.20i)14-s + 1.50·15-s + (−7.72 − 13.3i)16-s + (0.00903 − 0.0156i)17-s + (−3.87 + 6.70i)18-s + ⋯ |
L(s) = 1 | + (−0.969 − 1.67i)2-s + (−0.696 + 1.20i)3-s + (−1.38 + 2.39i)4-s + (−0.139 − 0.241i)5-s + 2.70·6-s + (0.480 + 0.876i)7-s + 3.41·8-s + (−0.470 − 0.815i)9-s + (−0.270 + 0.468i)10-s + (0.371 − 0.644i)11-s + (−1.92 − 3.33i)12-s + (1.00 − 1.65i)14-s + 0.389·15-s + (−1.93 − 3.34i)16-s + (0.00219 − 0.00379i)17-s + (−0.913 + 1.58i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6801412865\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6801412865\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.27 - 2.31i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.37 + 2.37i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.312 + 0.540i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 2.13i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.00903 + 0.0156i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.67 + 4.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.66 - 2.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.10T + 29T^{2} \) |
| 31 | \( 1 + (-2.58 + 4.47i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.94 - 3.37i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 - 0.158T + 43T^{2} \) |
| 47 | \( 1 + (0.732 + 1.26i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.41 - 2.44i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.25 - 7.37i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.86 + 8.42i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0205 + 0.0356i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 + (-6.26 + 10.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.40 - 7.63i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−9.718384755457385516142489427618, −9.167032829197165878264855854619, −8.594777815213715514711754065846, −7.76515056695074739809702238135, −6.15243432904348798563908838857, −4.82895617681598592296049748501, −4.42618281933357889935496427415, −3.29153714737358098810114837863, −2.31452874804411596166600730957, −0.76222203201200296542784882874,
0.77895949738542816012124188879, 1.67744385079222792761153488574, 4.24193461416245516888720470193, 5.14203738541288703153415147904, 6.19577450471968034717094888895, 6.66162627380980647204796775395, 7.37203887138811227163987373744, 7.81379453557553181412868332564, 8.656352347978506013994088798290, 9.657130792229731585496723679786