L(s) = 1 | + (−1.17 − 0.780i)2-s + (0.780 + 1.84i)4-s − 3.56·5-s − 4.05·7-s + (0.516 − 2.78i)8-s + 3·9-s + (4.19 + 2.78i)10-s + (0.848 + 3.20i)11-s + i·13-s + (4.78 + 3.16i)14-s + (−2.78 + 2.87i)16-s − 5.12i·17-s + (−3.53 − 2.34i)18-s + 4.34·19-s + (−2.78 − 6.55i)20-s + ⋯ |
L(s) = 1 | + (−0.833 − 0.552i)2-s + (0.390 + 0.920i)4-s − 1.59·5-s − 1.53·7-s + (0.182 − 0.983i)8-s + 9-s + (1.32 + 0.879i)10-s + (0.255 + 0.966i)11-s + 0.277i·13-s + (1.27 + 0.846i)14-s + (−0.695 + 0.718i)16-s − 1.24i·17-s + (−0.833 − 0.552i)18-s + 0.996·19-s + (−0.621 − 1.46i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.516539 - 0.253084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.516539 - 0.253084i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 + 0.780i)T \) |
| 11 | \( 1 + (-0.848 - 3.20i)T \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 17 | \( 1 + 5.12iT - 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.35iT - 23T^{2} \) |
| 29 | \( 1 + 6.68iT - 29T^{2} \) |
| 31 | \( 1 - 0.371iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 4.43iT - 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 + 6.41iT - 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4.05iT - 59T^{2} \) |
| 61 | \( 1 - 12.6iT - 61T^{2} \) |
| 67 | \( 1 + 14.0iT - 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 5.80iT - 73T^{2} \) |
| 79 | \( 1 - 1.32T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41500783697647008833213230504, −9.686400438989845610509199796501, −9.154097041819392289332952177369, −7.79130379032636118619467899849, −7.26254286134950311446086743322, −6.60631327788267020472453573134, −4.47372642840658501868297715355, −3.75913340845295272881227104628, −2.69143638999038295504169704289, −0.64130736857316607514653556348,
0.884172482720624392346134005117, 3.21009375289696978072793892441, 4.05481407292746852007568232088, 5.63139587426642080723628363688, 6.64743368130403801424402848541, 7.34091719735534430963220671981, 8.131604342215977888093047819449, 9.026964999478040029922521375969, 9.885389644561589447384293833734, 10.71164432639481590975167697964