| L(s) = 1 | + (−12.2 − 10.2i)2-s + (221. − 80.5i)3-s + (44.4 + 252. i)4-s + (−138. + 786. i)5-s + (−3.54e3 − 1.28e3i)6-s + (−456. − 791. i)7-s + (2.04e3 − 3.54e3i)8-s + (2.74e4 − 2.30e4i)9-s + (9.78e3 − 8.20e3i)10-s + (2.33e4 − 4.05e4i)11-s + (3.01e4 + 5.22e4i)12-s + (1.95e4 + 7.12e3i)13-s + (−2.53e3 + 1.43e4i)14-s + (3.26e4 + 1.85e5i)15-s + (−6.15e4 + 2.24e4i)16-s + (2.15e5 + 1.80e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (1.57 − 0.574i)3-s + (0.0868 + 0.492i)4-s + (−0.0991 + 0.562i)5-s + (−1.11 − 0.406i)6-s + (−0.0719 − 0.124i)7-s + (0.176 − 0.306i)8-s + (1.39 − 1.17i)9-s + (0.309 − 0.259i)10-s + (0.481 − 0.834i)11-s + (0.419 + 0.727i)12-s + (0.190 + 0.0691i)13-s + (−0.0176 + 0.100i)14-s + (0.166 + 0.944i)15-s + (−0.234 + 0.0855i)16-s + (0.626 + 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.14443 - 1.42875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14443 - 1.42875i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 + 10.2i)T \) |
| 19 | \( 1 + (-4.82e5 + 2.99e5i)T \) |
| good | 3 | \( 1 + (-221. + 80.5i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (138. - 786. i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (456. + 791. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.33e4 + 4.05e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.95e4 - 7.12e3i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-2.15e5 - 1.80e5i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (1.78e5 + 1.01e6i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-4.19e6 + 3.51e6i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (2.10e6 + 3.65e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 6.71e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.19e7 - 4.36e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (1.12e6 - 6.35e6i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (4.09e7 - 3.43e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (9.44e6 + 5.35e7i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-1.09e8 - 9.19e7i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-4.76e6 - 2.70e7i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (1.41e8 - 1.18e8i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (1.86e7 - 1.05e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-3.86e7 + 1.40e7i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (1.96e8 - 7.15e7i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-1.51e8 - 2.61e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.96e8 - 2.89e8i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (5.68e8 + 4.77e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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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
−14.02229007539567491958680201218, −13.06384067963036219267474837651, −11.62353959299588032150504296176, −10.11160691074103457788846438565, −8.867534362312178725444994168389, −7.963762821345656721220112544705, −6.72084771666197685037726504318, −3.63219456337516430606399630437, −2.65624994186056967078870977526, −1.10122623397469209957022171156,
1.52325817801020843012451553239, 3.32231988553534234614275247888, 4.94585973185222776427888783507, 7.22362387686280399093966345409, 8.404342563127675312893602555562, 9.284459549404786908755271093906, 10.17839073547048955085394942951, 12.24363644443703501450814831512, 13.81158261771083858007541938550, 14.61736072286830711452965905443
