L(s) = 1 | − 2.74·5-s + (1.70 + 2.02i)7-s + 0.418i·11-s + (−1.32 + 0.765i)13-s + (−1.95 − 3.38i)17-s + (−5.11 − 2.95i)19-s + 8.92i·23-s + 2.52·25-s + (−6.00 − 3.46i)29-s + (−3.05 − 1.76i)31-s + (−4.67 − 5.55i)35-s + (−4.54 + 7.87i)37-s + (1.06 + 1.84i)41-s + (−5.77 + 10.0i)43-s + (0.885 + 1.53i)47-s + ⋯ |
L(s) = 1 | − 1.22·5-s + (0.644 + 0.764i)7-s + 0.126i·11-s + (−0.367 + 0.212i)13-s + (−0.473 − 0.820i)17-s + (−1.17 − 0.678i)19-s + 1.86i·23-s + 0.505·25-s + (−1.11 − 0.643i)29-s + (−0.548 − 0.316i)31-s + (−0.790 − 0.938i)35-s + (−0.747 + 1.29i)37-s + (0.165 + 0.287i)41-s + (−0.881 + 1.52i)43-s + (0.129 + 0.223i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0769633 + 0.374988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0769633 + 0.374988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.70 - 2.02i)T \) |
good | 5 | \( 1 + 2.74T + 5T^{2} \) |
| 11 | \( 1 - 0.418iT - 11T^{2} \) |
| 13 | \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 8.92iT - 23T^{2} \) |
| 29 | \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.05 + 1.76i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.885 - 1.53i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.39 - 1.96i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.02 + 3.51i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 - 0.932i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.38 + 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.433 - 0.751i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.45 + 5.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.19060904882190786262472140945, −9.678162746002040236704751269145, −8.993180205784353430688342165965, −8.010490730018295630508770868271, −7.49079245777757521797916738910, −6.40924051998158788516099999284, −5.15059890149594082508094380682, −4.42684710844381742511743857538, −3.27647037187494549741319405043, −1.94629896074510304494899420411,
0.18643313208070900878616013519, 2.02564511928053227223895570396, 3.76192265165939203057006372938, 4.19370238582229635164400445341, 5.35230863317543422541862004899, 6.69913966750387357614267766411, 7.41259398551853847062364225847, 8.310005227032754538477047111482, 8.762384421781234894875972057963, 10.38377288295912945046550158201