L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.73 + 3i)5-s + (0.5 + 2.59i)7-s − 0.999i·8-s − 3.46i·10-s + (5.19 + 3i)11-s + (1.5 − 0.866i)13-s + (0.866 − 2.5i)14-s + (−0.5 + 0.866i)16-s + 1.73·17-s − 6.92i·19-s + (−1.73 + 2.99i)20-s + (−3 − 5.19i)22-s + (2.59 − 1.5i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.774 + 1.34i)5-s + (0.188 + 0.981i)7-s − 0.353i·8-s − 1.09i·10-s + (1.56 + 0.904i)11-s + (0.416 − 0.240i)13-s + (0.231 − 0.668i)14-s + (−0.125 + 0.216i)16-s + 0.420·17-s − 1.58i·19-s + (−0.387 + 0.670i)20-s + (−0.639 − 1.10i)22-s + (0.541 − 0.312i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.611056384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611056384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-1.73 - 3i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 - 1.5i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.73 - 3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.776893304525245095383779778441, −9.303512797403171969622179153803, −8.619276392291336127004730345009, −7.30828851573621878999333037715, −6.71512768256105797400079280194, −6.03348416015784148840690234468, −4.77835816922207746460533198812, −3.36659029838027701719742355204, −2.54229699463433584906642505901, −1.56603016969739666358938282100,
1.03389679832120234932848326167, 1.54563260237194862419751805862, 3.63228123526743971162753417572, 4.48761976047721948612286083240, 5.72949021301107508416401874465, 6.16339690870860772348726253597, 7.31111284455500055781266825658, 8.208974006031989910839314107417, 8.928712348632623465836752689149, 9.457738635814517281972945233069