L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.22 + 2.12i)5-s + (−1.62 − 2.09i)7-s − 0.999i·8-s − 2.44i·10-s + (−2.12 + 1.22i)13-s + (0.358 + 2.62i)14-s + (−0.5 + 0.866i)16-s + 4.89·17-s + 2.44i·19-s + (−1.22 + 2.12i)20-s + (5.19 − 3i)23-s + (−0.499 + 0.866i)25-s + 2.44·26-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.547 + 0.948i)5-s + (−0.612 − 0.790i)7-s − 0.353i·8-s − 0.774i·10-s + (−0.588 + 0.339i)13-s + (0.0958 + 0.700i)14-s + (−0.125 + 0.216i)16-s + 1.18·17-s + 0.561i·19-s + (−0.273 + 0.474i)20-s + (1.08 − 0.625i)23-s + (−0.0999 + 0.173i)25-s + 0.480·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.204797411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204797411\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.62 + 2.09i)T \) |
good | 5 | \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.19 - 3i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.44 - 4.24i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.44 - 4.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + (-6.12 - 10.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.6 - 6.12i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.22 - 2.12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (4.24 + 2.44i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.08549120738723288464243839644, −9.324581338091112024117990780216, −8.263071694927773198459106628047, −7.27552972182348425634624499830, −6.79381385742540732719089931001, −5.89657185803609177209833606446, −4.54020859609497368750067388052, −3.30666760498138646851810055640, −2.65084396659689621236777899250, −1.14431908871659870359626804471,
0.77468312068392539571631650601, 2.17068223303384374843705219824, 3.34598790844061005218052725199, 5.09548492771126262250083208737, 5.37023071941125775135933734386, 6.43058497195872797079405949838, 7.31265399779686809008845537071, 8.338883591924463528719155537049, 8.933264514859033904786671252110, 9.696004338752337054803424931271