L(s) = 1 | + (−1.5 − 0.866i)2-s − 3-s + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)5-s + (1.5 + 0.866i)6-s + (−2.5 − 0.866i)7-s + 1.73i·8-s − 2·9-s + 3·10-s + 5.19i·11-s + (−0.5 − 0.866i)12-s + (−1 − 3.46i)13-s + (3 + 3.46i)14-s + (1.5 − 0.866i)15-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + ⋯ |
L(s) = 1 | + (−1.06 − 0.612i)2-s − 0.577·3-s + (0.250 + 0.433i)4-s + (−0.670 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.944 − 0.327i)7-s + 0.612i·8-s − 0.666·9-s + 0.948·10-s + 1.56i·11-s + (−0.144 − 0.250i)12-s + (−0.277 − 0.960i)13-s + (0.801 + 0.925i)14-s + (0.387 − 0.223i)15-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.5 + 0.866i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 8.66iT - 67T^{2} \) |
| 71 | \( 1 + (-1.5 - 0.866i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.16371348442870399093902062507, −11.93305911788362475343826251061, −11.21711552819622516178340218458, −10.12907394878540675614875167831, −9.395539239652842239684726576896, −7.88654935467074101815542443262, −6.78328450407209927392109956700, −4.99121960582268942783276276215, −2.84908049331121638651684860775, 0,
3.72903192561893816413158387599, 5.86769882254247262044039795784, 6.76391275547223292903500878979, 8.434338348736293880930741680040, 8.791491402927148758584956909343, 10.25644898687776839236866277489, 11.45890042196068561769156560531, 12.42643715581397599566395255420, 13.63904160802417821624663628691