L(s) = 1 | + (3.11 − 3.11i)3-s + 4.53i·5-s + 0.745·7-s − 10.3i·9-s + (−0.0423 + 0.0423i)11-s − 8.30i·13-s + (14.0 + 14.0i)15-s + (15.2 − 15.2i)17-s + (25.1 − 25.1i)19-s + (2.31 − 2.31i)21-s − 8.64·23-s + 4.47·25-s + (−4.25 − 4.25i)27-s + (22.0 + 18.7i)29-s + (5.09 − 5.09i)31-s + ⋯ |
L(s) = 1 | + (1.03 − 1.03i)3-s + 0.906i·5-s + 0.106·7-s − 1.15i·9-s + (−0.00384 + 0.00384i)11-s − 0.638i·13-s + (0.939 + 0.939i)15-s + (0.897 − 0.897i)17-s + (1.32 − 1.32i)19-s + (0.110 − 0.110i)21-s − 0.375·23-s + 0.178·25-s + (−0.157 − 0.157i)27-s + (0.761 + 0.647i)29-s + (0.164 − 0.164i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.587200130\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.587200130\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (-22.0 - 18.7i)T \) |
good | 3 | \( 1 + (-3.11 + 3.11i)T - 9iT^{2} \) |
| 5 | \( 1 - 4.53iT - 25T^{2} \) |
| 7 | \( 1 - 0.745T + 49T^{2} \) |
| 11 | \( 1 + (0.0423 - 0.0423i)T - 121iT^{2} \) |
| 13 | \( 1 + 8.30iT - 169T^{2} \) |
| 17 | \( 1 + (-15.2 + 15.2i)T - 289iT^{2} \) |
| 19 | \( 1 + (-25.1 + 25.1i)T - 361iT^{2} \) |
| 23 | \( 1 + 8.64T + 529T^{2} \) |
| 31 | \( 1 + (-5.09 + 5.09i)T - 961iT^{2} \) |
| 37 | \( 1 + (-10.0 - 10.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-2.50 - 2.50i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-11.2 + 11.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-23.0 - 23.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 42.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 106.T + 3.48e3T^{2} \) |
| 61 | \( 1 + (42.3 - 42.3i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 - 75.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 71.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (73.7 + 73.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-78.7 + 78.7i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + 15.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + (22.8 - 22.8i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-42.8 - 42.8i)T + 9.40e3iT^{2} \) |
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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
−10.71449993171976433671315267479, −9.683152336359464310440153103528, −8.799454572005257985643166171268, −7.65799599196094910052530530877, −7.34849871077447631902720864440, −6.34053288455974651499066351842, −4.98537872809410524177200882137, −3.11125329880456511734890126257, −2.77851515113990855817547835520, −1.12062734956763570741676631182,
1.50902207644582822788117651922, 3.15732888606475692034809368439, 4.07441614061657918888186984362, 4.95937605624715149423023805819, 6.09935362146492079878747273579, 7.80399286106071513260568234316, 8.301136353589078853795203613392, 9.304917499812775072153746313341, 9.785100908681917741037123678004, 10.67448640679045861653017978782