L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 4·5-s + (2 − 1.73i)7-s + 0.999·8-s + (−2 + 3.46i)10-s + 2·11-s + (−3 + 5.19i)13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (2 + 3.46i)19-s + (−1.99 − 3.46i)20-s + (−1 + 1.73i)22-s − 23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.78·5-s + (0.755 − 0.654i)7-s + 0.353·8-s + (−0.632 + 1.09i)10-s + 0.603·11-s + (−0.832 + 1.44i)13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + (0.458 + 0.794i)19-s + (−0.447 − 0.774i)20-s + (−0.213 + 0.369i)22-s − 0.208·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.007714392\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.007714392\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - 4T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7 + 12.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.995617408436425078409034677619, −9.076376722519695400876553390122, −8.468943118728356199606270705692, −7.17038281345604630822505250321, −6.71963009136788546970115847276, −5.76338989330911575551128535889, −5.00479638376249526350044028376, −4.01585161716829614180808116174, −2.12780759358002515042350638291, −1.44416531294108733872477317606,
1.17233577440959354983042586402, 2.30419056717768278312420287508, 2.92671789224942069434235249108, 4.70902717534336579257394365724, 5.38075717366277091615707688109, 6.18342913169917189602604496992, 7.32889951669404321658555905866, 8.304317589688553146496621106430, 9.228154591062671527549276661427, 9.603751876424689916119410614117