L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (2 + 3.46i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.99 + 3.46i)10-s + (2 − 3.46i)11-s + (−0.499 − 0.866i)12-s − 4·13-s − 3.99·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (2 + 3.46i)19-s − 3.99·20-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.894 + 1.54i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.632 + 1.09i)10-s + (0.603 − 1.04i)11-s + (−0.144 − 0.249i)12-s − 1.10·13-s − 1.03·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (0.458 + 0.794i)19-s − 0.894·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.668907 + 1.34934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.668907 + 1.34934i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−11.92600478653742455092801157842, −11.18722997284546243693424283075, −10.09261000927074401530189068795, −9.566876486194536292856387732200, −8.118127987186609001252455802406, −6.90712420659255009124626991933, −6.19755467608876817418593479644, −5.34647480363494589389057159411, −3.78289372330496352835179715741, −2.66643210375454565960710431043,
1.16128895222006788874837597927, 2.37496962062192718155922833305, 4.55267023697914273424721480030, 5.08433840649501812970277510919, 6.27130054339859123027138020953, 7.53080043398771060766144080621, 8.976149509284075134878791999192, 9.501239565118286709381648977725, 10.47266445787759645887529501826, 11.97888609746625118435216627925