L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−1.5 − 2.59i)7-s − 3·8-s + (−1 − 1.73i)11-s + (−1 + 1.73i)13-s + (−1.5 + 2.59i)14-s + (0.500 + 0.866i)16-s + 4·17-s − 8·19-s + (−0.999 + 1.73i)22-s + (−1.5 + 2.59i)23-s + 1.99·26-s − 3·28-s + (−0.5 − 0.866i)29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.566 − 0.981i)7-s − 1.06·8-s + (−0.301 − 0.522i)11-s + (−0.277 + 0.480i)13-s + (−0.400 + 0.694i)14-s + (0.125 + 0.216i)16-s + 0.970·17-s − 1.83·19-s + (−0.213 + 0.369i)22-s + (−0.312 + 0.541i)23-s + 0.392·26-s − 0.566·28-s + (−0.0928 − 0.160i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.107326 + 0.608681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.107326 + 0.608681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.25808217177034498632273759870, −9.401494894108609174218915590874, −8.470043208626085190084086385835, −7.33722671160166898074880901519, −6.44951178544787877921830913337, −5.62169403098116426301867502272, −4.20189868866273516923037271542, −3.15278084172945382871504386796, −1.85580140037294899991126995904, −0.34066087748587836479031662230,
2.32785361445130168953501196763, 3.22906221062972436043939803167, 4.68635528239629710217673204112, 5.99612353950498737137371204888, 6.44108223885072474479195228379, 7.66024324288023754575593224679, 8.226927594636734235714688426153, 9.133939079565944715854203308098, 9.895183945586868705403190120068, 10.91405946265766374870851749679