L(s) = 1 | − 2-s + 4-s + (0.924 − 1.60i)5-s + (2.61 + 0.405i)7-s − 8-s + (−0.924 + 1.60i)10-s + (0.357 − 0.619i)11-s + (−2.81 + 2.25i)13-s + (−2.61 − 0.405i)14-s + 16-s − 4.31·17-s + (3.43 + 5.95i)19-s + (0.924 − 1.60i)20-s + (−0.357 + 0.619i)22-s + 7.16·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.413 − 0.716i)5-s + (0.988 + 0.153i)7-s − 0.353·8-s + (−0.292 + 0.506i)10-s + (0.107 − 0.186i)11-s + (−0.780 + 0.625i)13-s + (−0.698 − 0.108i)14-s + 0.250·16-s − 1.04·17-s + (0.788 + 1.36i)19-s + (0.206 − 0.358i)20-s + (−0.0762 + 0.132i)22-s + 1.49·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.454259779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454259779\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.405i)T \) |
| 13 | \( 1 + (2.81 - 2.25i)T \) |
good | 5 | \( 1 + (-0.924 + 1.60i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.357 + 0.619i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + (-3.43 - 5.95i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 + (-4.63 - 8.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.11 - 1.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + (2.23 + 3.86i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0979 - 0.169i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.60 - 7.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.80 + 6.58i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 + (4.31 + 7.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.96 + 12.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.50 - 13.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.989 + 1.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.37 + 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.71T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 + (-1.39 + 2.41i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.153150890523617568991280734567, −8.845079568487257784428470702815, −7.972449987648144039968427316102, −7.17633291132320817353998891450, −6.32612146225153550870956004659, −5.12142839290069357849592043392, −4.78307664472960531079572897108, −3.27016090880937272270066531023, −1.96610405212309024831764852596, −1.18513438817106524617286437141,
0.823384135428319613846666851292, 2.30314115480188407843592789313, 2.87810972838036403759172989540, 4.49760997082035625070632012042, 5.18563819652685280193532728090, 6.40274127339661844729004842699, 7.04582361279093333407710733464, 7.73043542189118470584550547422, 8.581422304196612744147236138162, 9.360905275319818987656569240283