| L(s) = 1 | + (5.61 + 24.6i)2-s + (39.7 + 36.8i)3-s + (−112. + 54.2i)4-s + (1.15e3 − 173. i)5-s + (−683. + 1.18e3i)6-s + (−903. − 1.56e3i)7-s + (6.08e3 + 7.63e3i)8-s + (−1.25e3 − 1.66e4i)9-s + (1.07e4 + 2.73e4i)10-s + (4.76e4 + 2.29e4i)11-s + (−6.47e3 − 1.99e3i)12-s + (3.47e4 − 8.85e4i)13-s + (3.34e4 − 3.10e4i)14-s + (5.20e4 + 3.55e4i)15-s + (−1.93e5 + 2.42e5i)16-s + (6.33e5 + 9.54e4i)17-s + ⋯ |
| L(s) = 1 | + (0.248 + 1.08i)2-s + (0.283 + 0.262i)3-s + (−0.219 + 0.105i)4-s + (0.822 − 0.124i)5-s + (−0.215 + 0.373i)6-s + (−0.142 − 0.246i)7-s + (0.525 + 0.659i)8-s + (−0.0635 − 0.848i)9-s + (0.339 + 0.864i)10-s + (0.981 + 0.472i)11-s + (−0.0901 − 0.0277i)12-s + (0.337 − 0.860i)13-s + (0.232 − 0.215i)14-s + (0.265 + 0.181i)15-s + (−0.738 + 0.926i)16-s + (1.83 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.55818 + 2.12371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55818 + 2.12371i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.23e7 + 1.48e6i)T \) |
| good | 2 | \( 1 + (-5.61 - 24.6i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-39.7 - 36.8i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-1.15e3 + 173. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (903. + 1.56e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-4.76e4 - 2.29e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-3.47e4 + 8.85e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-6.33e5 - 9.54e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (4.27e3 - 5.70e4i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (2.77e5 - 1.88e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (3.11e6 - 2.89e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (3.54e6 + 1.09e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (3.23e6 - 5.60e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-6.57e6 - 2.88e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-4.90e7 + 2.36e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-1.10e7 - 2.80e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-6.27e7 + 7.86e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (1.56e8 - 4.81e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-1.05e7 + 1.40e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.03e8 + 7.02e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (-1.09e8 + 2.79e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (2.05e8 + 3.55e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.69e8 - 2.50e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-5.84e8 - 5.42e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-1.05e9 - 5.10e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.58986470955338610423284880572, −13.48143576097041938448315792045, −12.06917657245001756100866134953, −10.30996057049205572106504729481, −9.230444960612443454548795996340, −7.74506287566320399058633749721, −6.38913010906521857825601204952, −5.44841751598424121271889345117, −3.60230947461561658573728889324, −1.42245510039014663956220923861,
1.33701593525514695028599100013, 2.36326596034567383816645767193, 3.79533958785828072527701365836, 5.76160683902237066722314169099, 7.35501662524467588955096033182, 9.122401639776959937120101083149, 10.24054605830417204233539649443, 11.42949877212977173167327562735, 12.42724880500525950352830530983, 13.70214614825572369444042134021
