| L(s) = 1 | + (6.12 − 5.14i)2-s + (−18.0 − 6.57i)3-s + (11.1 − 63.0i)4-s + (22.8 + 129. i)5-s + (−144. + 52.5i)6-s + (446. − 773. i)7-s + (−256. − 443. i)8-s + (−1.39e3 − 1.16e3i)9-s + (805. + 675. i)10-s + (−1.82e3 − 3.15e3i)11-s + (−614. + 1.06e3i)12-s + (−1.18e4 + 4.29e3i)13-s + (−1.24e3 − 7.03e3i)14-s + (438. − 2.48e3i)15-s + (−3.84e3 − 1.40e3i)16-s + (−1.33e3 + 1.12e3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.386 − 0.140i)3-s + (0.0868 − 0.492i)4-s + (0.0816 + 0.463i)5-s + (−0.272 + 0.0993i)6-s + (0.491 − 0.852i)7-s + (−0.176 − 0.306i)8-s + (−0.636 − 0.534i)9-s + (0.254 + 0.213i)10-s + (−0.412 − 0.714i)11-s + (−0.102 + 0.177i)12-s + (−1.49 + 0.542i)13-s + (−0.120 − 0.685i)14-s + (0.0335 − 0.190i)15-s + (−0.234 − 0.0855i)16-s + (−0.0660 + 0.0553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.273519 - 1.25818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.273519 - 1.25818i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.12 + 5.14i)T \) |
| 19 | \( 1 + (7.93e3 + 2.88e4i)T \) |
| good | 3 | \( 1 + (18.0 + 6.57i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-22.8 - 129. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-446. + 773. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.82e3 + 3.15e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.18e4 - 4.29e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.33e3 - 1.12e3i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 23 | \( 1 + (5.59e3 - 3.17e4i)T + (-3.19e9 - 1.16e9i)T^{2} \) |
| 29 | \( 1 + (1.80e5 + 1.51e5i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-1.18e5 + 2.04e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.97e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.69e5 - 1.34e5i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-8.61e4 - 4.88e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-5.59e5 - 4.69e5i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-1.36e5 + 7.74e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (4.47e5 - 3.75e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.43e5 + 8.13e5i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (2.87e6 + 2.41e6i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-5.09e5 - 2.89e6i)T + (-8.54e12 + 3.11e12i)T^{2} \) |
| 73 | \( 1 + (1.07e6 + 3.92e5i)T + (8.46e12 + 7.10e12i)T^{2} \) |
| 79 | \( 1 + (1.50e6 + 5.46e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-3.91e6 + 6.78e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (6.19e6 - 2.25e6i)T + (3.38e13 - 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-2.59e6 + 2.17e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−14.22009596382111057693324327719, −13.09948702561676453085748941498, −11.63089627354686310208662864216, −10.96292248754378648983408466238, −9.517806910892984410265651221563, −7.50707341464263622217631761550, −6.06141776530204576101172865759, −4.47545202085557461146686031923, −2.65376958903807396599161542373, −0.47167926133499911604405426996,
2.42548308127465370957548050172, 4.84346253814485538275314844071, 5.57427520035754587698675769125, 7.49652973221860236038371253862, 8.781825512307755304900847168787, 10.49529500441567696856270403076, 12.04491137229791188488251163566, 12.71365207323593950769262832156, 14.37606200659569255303928095647, 15.13258739830364596178993326429
