| L(s) = 1 | + (6.12 − 5.14i)2-s + (69.8 + 25.4i)3-s + (11.1 − 63.0i)4-s + (24.1 + 137. i)5-s + (558. − 203. i)6-s + (520. − 901. i)7-s + (−256. − 443. i)8-s + (2.55e3 + 2.14e3i)9-s + (853. + 715. i)10-s + (1.67e3 + 2.89e3i)11-s + (2.37e3 − 4.11e3i)12-s + (−232. + 84.6i)13-s + (−1.44e3 − 8.20e3i)14-s + (−1.79e3 + 1.01e4i)15-s + (−3.84e3 − 1.40e3i)16-s + (−3.34 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (1.49 + 0.543i)3-s + (0.0868 − 0.492i)4-s + (0.0864 + 0.490i)5-s + (1.05 − 0.384i)6-s + (0.573 − 0.993i)7-s + (−0.176 − 0.306i)8-s + (1.16 + 0.979i)9-s + (0.269 + 0.226i)10-s + (0.378 + 0.655i)11-s + (0.397 − 0.687i)12-s + (−0.0293 + 0.0106i)13-s + (−0.140 − 0.798i)14-s + (−0.137 + 0.779i)15-s + (−0.234 − 0.0855i)16-s + (−0.000165 + 0.000138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.91677 - 0.430861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.91677 - 0.430861i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (1.10e4 - 2.77e4i)T \) |
| good | 3 | \( 1 + (-69.8 - 25.4i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-24.1 - 137. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-520. + 901. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.67e3 - 2.89e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (232. - 84.6i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (3.34 - 2.80i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.34e4 + 7.62e4i)T + (-3.19e9 - 1.16e9i)T^{2} \) |
| 29 | \( 1 + (1.06e5 + 8.90e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.00e5 - 1.74e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.11e5 + 2.59e5i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-9.68e4 - 5.49e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (4.42e4 + 3.71e4i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + (1.31e5 - 7.47e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-2.99e5 + 2.51e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-3.49e5 + 1.98e6i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (8.04e4 + 6.74e4i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (5.52e3 + 3.13e4i)T + (-8.54e12 + 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-3.03e6 - 1.10e6i)T + (8.46e12 + 7.10e12i)T^{2} \) |
| 79 | \( 1 + (5.26e5 + 1.91e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (1.95e6 - 3.39e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.23e7 - 4.49e6i)T + (3.38e13 - 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-7.74e6 + 6.49e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−14.39955870692750493068007447704, −14.02566357144414177094282908499, −12.58332270497896536875794123498, −10.77803995950164826378036564391, −9.943404723741923550599876000221, −8.481854319192545132048909949298, −7.02963760415570876568156135075, −4.53006531394927284878570763634, −3.43438482275011048208606662201, −1.88432192859352716598975875955,
1.87238211013987603688720882151, 3.38604959012278820222348491925, 5.32936427492502815234482765083, 7.16534688437095771962871026958, 8.528498827206880979639432163214, 9.059359110758084492316106460781, 11.57070387804451721802961859123, 12.89255445717711151803874406432, 13.68930965726769956813885951805, 14.78110970266088449391633899918
