| L(s) = 1 | + (−5.53 − 9.58i)3-s + (654. + 378. i)5-s + (680. − 6.31e3i)7-s + (9.78e3 − 1.69e4i)9-s + (5.09e4 − 2.94e4i)11-s + 1.56e4i·13-s − 8.37e3i·15-s + (−2.64e5 + 1.52e5i)17-s + (2.86e4 − 4.96e4i)19-s + (−6.43e4 + 2.84e4i)21-s + (−5.22e5 − 3.01e5i)23-s + (−6.90e5 − 1.19e6i)25-s − 4.34e5·27-s + 2.18e6·29-s + (2.56e6 + 4.45e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.0394 − 0.0683i)3-s + (0.468 + 0.270i)5-s + (0.107 − 0.994i)7-s + (0.496 − 0.860i)9-s + (1.04 − 0.605i)11-s + 0.151i·13-s − 0.0426i·15-s + (−0.768 + 0.443i)17-s + (0.0504 − 0.0873i)19-s + (−0.0721 + 0.0319i)21-s + (−0.389 − 0.224i)23-s + (−0.353 − 0.612i)25-s − 0.157·27-s + 0.572·29-s + (0.499 + 0.865i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.094692058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094692058\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-680. + 6.31e3i)T \) |
| good | 3 | \( 1 + (5.53 + 9.58i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-654. - 378. i)T + (9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-5.09e4 + 2.94e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.56e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (2.64e5 - 1.52e5i)T + (5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.86e4 + 4.96e4i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (5.22e5 + 3.01e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 2.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.56e6 - 4.45e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (4.31e5 - 7.47e5i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 9.62e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 2.43e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (6.97e6 - 1.20e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (5.35e7 + 9.26e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.51e7 - 4.35e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.06e7 + 4.65e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.98e8 + 1.14e8i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.20e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.01e7 + 1.16e7i)T + (2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.09e8 - 1.20e8i)T + (5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (8.63e8 + 4.98e8i)T + (1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.37e9iT - 7.60e17T^{2} \) |
| 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
−11.50396264551047484129035590294, −10.43980359660152573383058000677, −9.524594845134622938911380473310, −8.347857564956033740435277052643, −6.81829222179043289427689038257, −6.30285256959486591750088731601, −4.45653411627052080549914415874, −3.45972168417724575426496344797, −1.67380095187651998463204947825, −0.53838456868503516162744845427,
1.45324187792610907815949449364, 2.44534389931907804650706224365, 4.28435413893083822051253713999, 5.35201283916789636974643050089, 6.53577093512166212299314378545, 7.87009420216223238786980935480, 9.112722809181016588772770755094, 9.829909484972323700596415832922, 11.19709728039296027288450552666, 12.14067774256567308451660862181
