| L(s) = 1 | + (54.1 + 45.4i)3-s + (−67.9 + 24.7i)5-s + (−564. − 977. i)7-s + (486. + 2.76e3i)9-s + (−1.08e3 + 1.88e3i)11-s + (−1.17e4 + 9.85e3i)13-s + (−4.79e3 − 1.74e3i)15-s + (−5.83e3 + 3.30e4i)17-s + (−1.98e4 − 2.23e4i)19-s + (1.38e4 − 7.85e4i)21-s + (3.17e4 + 1.15e4i)23-s + (−5.58e4 + 4.68e4i)25-s + (−2.17e4 + 3.76e4i)27-s + (−7.59e3 − 4.30e4i)29-s + (−6.95e4 − 1.20e5i)31-s + ⋯ |
| L(s) = 1 | + (1.15 + 0.970i)3-s + (−0.242 + 0.0884i)5-s + (−0.621 − 1.07i)7-s + (0.222 + 1.26i)9-s + (−0.246 + 0.426i)11-s + (−1.48 + 1.24i)13-s + (−0.366 − 0.133i)15-s + (−0.288 + 1.63i)17-s + (−0.664 − 0.746i)19-s + (0.326 − 1.85i)21-s + (0.543 + 0.197i)23-s + (−0.714 + 0.599i)25-s + (−0.212 + 0.368i)27-s + (−0.0577 − 0.327i)29-s + (−0.419 − 0.726i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0886141 + 1.09833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0886141 + 1.09833i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.98e4 + 2.23e4i)T \) |
| good | 3 | \( 1 + (-54.1 - 45.4i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (67.9 - 24.7i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (564. + 977. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.08e3 - 1.88e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.17e4 - 9.85e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (5.83e3 - 3.30e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-3.17e4 - 1.15e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (7.59e3 + 4.30e4i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (6.95e4 + 1.20e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.81e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.93e5 + 2.45e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (6.46e5 - 2.35e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-2.13e5 - 1.20e6i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-7.52e5 - 2.73e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-3.53e5 + 2.00e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (4.26e5 + 1.55e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-3.03e5 - 1.71e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-4.70e6 + 1.71e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-3.19e5 - 2.68e5i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-4.08e5 - 3.42e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-2.03e6 - 3.52e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.28e6 - 6.95e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (5.50e5 - 3.11e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−13.74466018458981589666933594427, −12.76657263682145106474720295078, −11.08428443678519683335556122110, −9.945776021057537152134444492436, −9.335458959447003902020794161879, −7.940817581550555892895338237908, −6.80490678075366724260660223682, −4.51716131892213184696455039342, −3.77350606761829377197449689130, −2.25878217648360651624691999611,
0.29305755858513870488094426265, 2.33763324716257274382153810013, 3.06839608580371532407657002332, 5.32122686183244302366733408756, 6.91196313620177306994755218513, 7.967989381984372216672184608204, 8.874217229108318403507080972330, 10.00975094447974898844805538320, 11.91596763734700956265356639648, 12.65443462774992134263377900755
