| L(s) = 1 | + (2.77 − 15.7i)2-s + (−127. − 106. i)3-s + (−240. − 87.5i)4-s + (2.13e3 − 778. i)5-s + (−2.03e3 + 1.71e3i)6-s + (−5.33e3 − 9.24e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (1.38e3 + 7.85e3i)9-s + (−6.32e3 − 3.58e4i)10-s + (3.76e4 − 6.52e4i)11-s + (2.12e4 + 3.68e4i)12-s + (1.06e5 − 8.92e4i)13-s + (−1.60e5 + 5.84e4i)14-s + (−3.55e5 − 1.29e5i)15-s + (5.02e4 + 4.21e4i)16-s + (−6.44e4 + 3.65e5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.908 − 0.761i)3-s + (−0.469 − 0.171i)4-s + (1.53 − 0.556i)5-s + (−0.642 + 0.538i)6-s + (−0.840 − 1.45i)7-s + (−0.176 + 0.306i)8-s + (0.0703 + 0.398i)9-s + (−0.199 − 1.13i)10-s + (0.775 − 1.34i)11-s + (0.296 + 0.513i)12-s + (1.03 − 0.866i)13-s + (−1.11 + 0.406i)14-s + (−1.81 − 0.660i)15-s + (0.191 + 0.160i)16-s + (−0.187 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.435928 + 1.48152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435928 + 1.48152i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.77 + 15.7i)T \) |
| 19 | \( 1 + (5.01e5 - 2.67e5i)T \) |
| good | 3 | \( 1 + (127. + 106. i)T + (3.41e3 + 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-2.13e3 + 778. i)T + (1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (5.33e3 + 9.24e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.76e4 + 6.52e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.06e5 + 8.92e4i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (6.44e4 - 3.65e5i)T + (-1.11e11 - 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-1.07e5 - 3.93e4i)T + (1.37e12 + 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-5.50e5 - 3.12e6i)T + (-1.36e13 + 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-6.83e5 - 1.18e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.10e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.00e6 + 1.68e6i)T + (5.68e13 + 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-2.54e7 + 9.27e6i)T + (3.85e14 - 3.23e14i)T^{2} \) |
| 47 | \( 1 + (7.90e6 + 4.48e7i)T + (-1.05e15 + 3.82e14i)T^{2} \) |
| 53 | \( 1 + (5.50e7 + 2.00e7i)T + (2.52e15 + 2.12e15i)T^{2} \) |
| 59 | \( 1 + (1.47e7 - 8.35e7i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (-3.72e7 - 1.35e7i)T + (8.95e15 + 7.51e15i)T^{2} \) |
| 67 | \( 1 + (-2.24e7 - 1.27e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (-1.38e8 + 5.02e7i)T + (3.51e16 - 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-1.23e8 - 1.03e8i)T + (1.02e16 + 5.79e16i)T^{2} \) |
| 79 | \( 1 + (2.76e8 + 2.31e8i)T + (2.08e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-1.10e8 - 1.91e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-8.00e7 + 6.71e7i)T + (6.08e16 - 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-2.17e8 + 1.23e9i)T + (-7.14e17 - 2.60e17i)T^{2} \) |
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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
−13.18539808335385418466853658881, −12.76849322003431540878284711675, −11.00213525156024409193482575997, −10.23172679383942744355501194361, −8.740530668847641565983796023512, −6.42090093787818472805195375113, −5.83292623425306539641708085221, −3.66581399059360847274303817131, −1.35795439336774081130750684971, −0.66945465243374766062188845645,
2.31329463119965824630163560396, 4.65334307796290768344897868487, 6.03542987654367025923821844371, 6.48785968950467414705619534168, 9.286014090499725132206209042703, 9.662632472085389830521207864979, 11.29742518796442593854358477985, 12.71175853763681904935526769823, 13.99877556422249038718726584613, 15.21547906341889576327661316319
