| L(s) = 1 | + (13.3 + 8.79i)2-s + (−67.4 + 67.4i)3-s + (101. + 235. i)4-s + (191. − 191. i)5-s + (−1.49e3 + 309. i)6-s − 3.93e3·7-s + (−709. + 4.03e3i)8-s − 2.54e3i·9-s + (4.24e3 − 877. i)10-s + (−868. − 868. i)11-s + (−2.27e4 − 9.01e3i)12-s + (2.82e4 + 2.82e4i)13-s + (−5.26e4 − 3.45e4i)14-s + 2.58e4i·15-s + (−4.49e4 + 4.76e4i)16-s + 5.60e4·17-s + ⋯ |
| L(s) = 1 | + (0.835 + 0.549i)2-s + (−0.833 + 0.833i)3-s + (0.396 + 0.918i)4-s + (0.306 − 0.306i)5-s + (−1.15 + 0.238i)6-s − 1.63·7-s + (−0.173 + 0.984i)8-s − 0.388i·9-s + (0.424 − 0.0877i)10-s + (−0.0593 − 0.0593i)11-s + (−1.09 − 0.434i)12-s + (0.988 + 0.988i)13-s + (−1.36 − 0.900i)14-s + 0.511i·15-s + (−0.685 + 0.727i)16-s + 0.671·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.272957 + 1.48733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272957 + 1.48733i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-13.3 - 8.79i)T \) |
| good | 3 | \( 1 + (67.4 - 67.4i)T - 6.56e3iT^{2} \) |
| 5 | \( 1 + (-191. + 191. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + 3.93e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (868. + 868. i)T + 2.14e8iT^{2} \) |
| 13 | \( 1 + (-2.82e4 - 2.82e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 - 5.60e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + (3.66e4 - 3.66e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 - 4.69e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (2.98e5 + 2.98e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 - 1.42e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-8.55e5 + 8.55e5i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + 3.48e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (2.28e6 + 2.28e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 - 2.66e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + (2.55e6 - 2.55e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + (-5.04e6 - 5.04e6i)T + 1.46e14iT^{2} \) |
| 61 | \( 1 + (5.51e6 + 5.51e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 + (-2.78e7 + 2.78e7i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 - 1.70e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.86e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.23e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.20e7 + 2.20e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 9.93e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.23e8T + 7.83e15T^{2} \) |
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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
−16.98033992871536328322112670030, −16.41450638260711692764884788204, −15.50529155904618100971257819818, −13.66511426936085595238060809695, −12.51378252481433171303335022662, −10.92790022987691206370514401934, −9.228148787619065237903792733333, −6.64281438868079611698737907363, −5.38882170895308489684770052740, −3.63203363507200624867543273151,
0.72697589560932424591790809837, 3.13899294498476468281161493196, 5.83080061903463414288456205242, 6.71431824728193304208999395831, 9.865194811640633710124920308585, 11.25873911782547452256057162484, 12.81017070183811467307146875413, 13.18747025331062558461502672272, 15.15215728217314638987377565241, 16.59053288709298034399200883472
