| L(s) = 1 | + (−1.14 + 3.83i)2-s + (9.44 − 1.66i)3-s + (−13.3 − 8.78i)4-s + (−12.8 + 10.7i)5-s + (−4.44 + 38.1i)6-s + (−66.0 − 38.1i)7-s + (48.9 − 41.1i)8-s + (10.3 − 3.76i)9-s + (−26.5 − 61.5i)10-s + (−169. + 97.7i)11-s + (−140. − 60.6i)12-s + (−29.5 + 167. i)13-s + (221. − 209. i)14-s + (−103. + 123. i)15-s + (101. + 234. i)16-s + (11.9 + 4.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.286 + 0.958i)2-s + (1.04 − 0.185i)3-s + (−0.835 − 0.548i)4-s + (−0.513 + 0.430i)5-s + (−0.123 + 1.05i)6-s + (−1.34 − 0.778i)7-s + (0.765 − 0.643i)8-s + (0.127 − 0.0465i)9-s + (−0.265 − 0.615i)10-s + (−1.39 + 0.807i)11-s + (−0.978 − 0.421i)12-s + (−0.175 + 0.992i)13-s + (1.13 − 1.06i)14-s + (−0.459 + 0.547i)15-s + (0.397 + 0.917i)16-s + (0.0413 + 0.0150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0697982 - 0.208636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0697982 - 0.208636i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.14 - 3.83i)T \) |
| 19 | \( 1 + (194. + 304. i)T \) |
| good | 3 | \( 1 + (-9.44 + 1.66i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (12.8 - 10.7i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (66.0 + 38.1i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (169. - 97.7i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (29.5 - 167. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-11.9 - 4.34i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-656. + 781. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-721. + 262. i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-256. - 147. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.28e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-387. - 2.19e3i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (66.3 + 79.0i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-28.9 - 79.5i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (2.93e3 + 2.46e3i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (1.80e3 - 4.96e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-866. - 727. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-167. - 460. i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (2.38e3 + 2.84e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-741. - 4.20e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-637. + 112. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (8.93e3 + 5.15e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-984. + 5.58e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (1.30e3 + 473. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−14.58390106345912699161810956321, −13.48122474724765098114990776776, −12.86810581632496269802589529476, −10.68143318918556139246379689564, −9.661434383851578057906657671531, −8.556652061517733504411442027487, −7.35172171847669243345025700070, −6.70571063512362056062810815914, −4.55235764066900023904997525508, −2.90507030043268471624703088296,
0.10316996219985525823830965804, 2.76179969940873176797614526270, 3.43744347262971785987979798723, 5.47146132408499571642915946384, 7.918611961124336473298285406081, 8.641344301483992290849643760453, 9.631099028077337892838315056210, 10.69920612630757318682994271064, 12.24683541649588647837669824587, 12.94381855118423596894269774946
