L(s) = 1 | + (−5.49 + 5.49i)3-s + (−4.66 − 4.66i)5-s − 24.8i·7-s − 33.4i·9-s + (−22.3 − 22.3i)11-s + (−11.2 + 11.2i)13-s + 51.2·15-s − 88.4·17-s + (−37.8 + 37.8i)19-s + (136. + 136. i)21-s + 48.1i·23-s − 81.4i·25-s + (35.2 + 35.2i)27-s + (10.4 − 10.4i)29-s + 96.9·31-s + ⋯ |
L(s) = 1 | + (−1.05 + 1.05i)3-s + (−0.417 − 0.417i)5-s − 1.34i·7-s − 1.23i·9-s + (−0.612 − 0.612i)11-s + (−0.240 + 0.240i)13-s + 0.882·15-s − 1.26·17-s + (−0.456 + 0.456i)19-s + (1.42 + 1.42i)21-s + 0.436i·23-s − 0.651i·25-s + (0.251 + 0.251i)27-s + (0.0668 − 0.0668i)29-s + 0.561·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.141744 - 0.259805i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.141744 - 0.259805i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (5.49 - 5.49i)T - 27iT^{2} \) |
| 5 | \( 1 + (4.66 + 4.66i)T + 125iT^{2} \) |
| 7 | \( 1 + 24.8iT - 343T^{2} \) |
| 11 | \( 1 + (22.3 + 22.3i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (11.2 - 11.2i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 88.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + (37.8 - 37.8i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 48.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-10.4 + 10.4i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 96.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + (163. + 163. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 360. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-100. - 100. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (175. + 175. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (405. + 405. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-664. + 664. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-107. + 107. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 215. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 668. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 822.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (326. - 326. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 262. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 150.T + 9.12e5T^{2} \) |
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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
−14.06582958568053051250019358591, −12.85686257611817355518347225257, −11.41340954701605657510550209120, −10.75182056216007347660158727906, −9.770415910267995599806886660579, −8.136041754213619151200383658665, −6.50278905980661357821655611895, −4.91204636899722761909872859586, −3.98641014341787647890747659331, −0.21883519007537294758173720720,
2.32315365300270425872717217964, 5.07392092569528286742347654677, 6.33520672240567997980768824371, 7.36715510546635874900084694884, 8.816889072631537140391828041629, 10.63372963680282454274324311090, 11.67796291827843450976752553824, 12.41758293544252354880770483530, 13.30310534410837565698214089828, 15.06024120481589020645124064270