L(s) = 1 | + (−0.746 − 1.29i)3-s + (0.631 − 1.09i)5-s + (2.25 + 1.37i)7-s + (0.385 − 0.667i)9-s + (−0.5 − 0.866i)11-s + 6.62·13-s − 1.88·15-s + (0.512 + 0.888i)17-s + (−1.05 + 1.82i)19-s + (0.0936 − 3.94i)21-s + (−0.923 + 1.60i)23-s + (1.70 + 2.94i)25-s − 5.63·27-s + 4.86·29-s + (0.804 + 1.39i)31-s + ⋯ |
L(s) = 1 | + (−0.430 − 0.746i)3-s + (0.282 − 0.489i)5-s + (0.853 + 0.520i)7-s + (0.128 − 0.222i)9-s + (−0.150 − 0.261i)11-s + 1.83·13-s − 0.487·15-s + (0.124 + 0.215i)17-s + (−0.241 + 0.417i)19-s + (0.0204 − 0.861i)21-s + (−0.192 + 0.333i)23-s + (0.340 + 0.589i)25-s − 1.08·27-s + 0.903·29-s + (0.144 + 0.250i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.832900928\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.832900928\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.25 - 1.37i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.746 + 1.29i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.631 + 1.09i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 6.62T + 13T^{2} \) |
| 17 | \( 1 + (-0.512 - 0.888i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.05 - 1.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.923 - 1.60i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 + (-0.804 - 1.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.08 + 1.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + (-2.51 + 4.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.26 + 2.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.77 + 3.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.47 + 2.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.92 + 8.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + (7.61 + 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.15 - 7.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 + (3.31 - 5.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.10T + 97T^{2} \) |
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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
−9.397813920962261667384519477986, −8.634728232302909732380422591600, −8.091652535304750684983331369766, −7.06141437706680899389547801641, −6.01748465999923177033163217761, −5.70155979867394150400835094366, −4.50826439664054478096276768690, −3.40178587071073102551565292984, −1.80450407424891320991010671652, −1.06697862308045917145829729113,
1.25458792103916249558726660576, 2.66373879834520898789361393920, 4.04377394295757347768152283731, 4.55894391861294736515444390069, 5.61474944408980174077671602803, 6.40978661776023042850439532370, 7.39373593190276029915623093055, 8.276419138422282114141590464992, 9.035868185791206165795834351965, 10.27659965886258945854017098277