L(s) = 1 | − 2-s + (1.37 − 1.05i)3-s + 4-s + 3.92i·5-s + (−1.37 + 1.05i)6-s + i·7-s − 8-s + (0.775 − 2.89i)9-s − 3.92i·10-s + (1.42 + 2.99i)11-s + (1.37 − 1.05i)12-s − 0.109i·13-s − i·14-s + (4.14 + 5.39i)15-s + 16-s − 6.97·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.793 − 0.608i)3-s + 0.5·4-s + 1.75i·5-s + (−0.560 + 0.430i)6-s + 0.377i·7-s − 0.353·8-s + (0.258 − 0.965i)9-s − 1.24i·10-s + (0.430 + 0.902i)11-s + (0.396 − 0.304i)12-s − 0.0302i·13-s − 0.267i·14-s + (1.06 + 1.39i)15-s + 0.250·16-s − 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06589 + 0.653443i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06589 + 0.653443i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-1.42 - 2.99i)T \) |
good | 5 | \( 1 - 3.92iT - 5T^{2} \) |
| 13 | \( 1 + 0.109iT - 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 19 | \( 1 - 5.78iT - 19T^{2} \) |
| 23 | \( 1 - 0.938iT - 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 1.44iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 7.60iT - 59T^{2} \) |
| 61 | \( 1 + 6.35iT - 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 - 4.39iT - 71T^{2} \) |
| 73 | \( 1 - 3.31iT - 73T^{2} \) |
| 79 | \( 1 + 0.361iT - 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 - 0.722iT - 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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
−11.10100062465345080830420056510, −10.15351042438030445906529347604, −9.470030156671747162927391147157, −8.410999689981672027953446234417, −7.53397322521884640581368498145, −6.74073919816005306847516763988, −6.23480948544139299712230308799, −4.00666810879765836097574170679, −2.75845745771989833176314290614, −1.99381967249968696485408618033,
0.914638940317646164890418469371, 2.55686605449881183096465074991, 4.19873033015271524613059330128, 4.83222932299571880400342738382, 6.29232061044267350968181987785, 7.64182644704391812236607161865, 8.679458995771583348532516721360, 8.861675733865629220617833502167, 9.626683057499041793182432623913, 10.80699941982024120213924739340