L(s) = 1 | + (1.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (1.5 + 0.866i)6-s + (3 + 1.73i)7-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + 12-s + 6·14-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s + 1.73i·18-s + (−3 − 1.73i)19-s + 3.46i·21-s + (−3 + 5.19i)22-s + ⋯ |
L(s) = 1 | + (1.06 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.612 + 0.353i)6-s + (1.13 + 0.654i)7-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + 0.288·12-s + 1.60·14-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s + 0.408i·18-s + (−0.688 − 0.397i)19-s + 0.755i·21-s + (−0.639 + 1.10i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.73192 + 0.368578i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.73192 + 0.368578i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (9 + 5.19i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9 - 5.19i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 1.73i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.22136490548050238700341296757, −10.32681603265505707390544823560, −9.222598148460125475631790069369, −8.269346925509371819170113405755, −7.50744271230282514883165145012, −5.74946229650646569566044859803, −4.95959133519996698833543753451, −4.38415144156080688565268646400, −2.94476769164125798464061857377, −2.16353542090953059532271659962,
1.39413030837277099293824169179, 3.19438602006445547739978125456, 4.29942459972492733596896472881, 5.25260540768434304842781425456, 6.11226896186732016946064861397, 7.17973129640586988754649034104, 7.937053758127701116969054986705, 8.707112751648232392472390546857, 10.28342073372220223343490437356, 10.81978271953389441699002435642