L(s) = 1 | + (−1.33 + 3.77i)2-s + (−4.88 + 0.861i)3-s + (−12.4 − 10.0i)4-s + (−29.7 + 24.9i)5-s + (3.26 − 19.5i)6-s + (20.5 + 11.8i)7-s + (54.5 − 33.5i)8-s + (−52.9 + 19.2i)9-s + (−54.5 − 145. i)10-s + (101. − 58.5i)11-s + (69.4 + 38.4i)12-s + (52.2 − 296. i)13-s + (−72.0 + 61.5i)14-s + (124. − 147. i)15-s + (53.8 + 250. i)16-s + (28.4 + 10.3i)17-s + ⋯ |
L(s) = 1 | + (−0.333 + 0.942i)2-s + (−0.543 + 0.0957i)3-s + (−0.777 − 0.628i)4-s + (−1.19 + 0.999i)5-s + (0.0906 − 0.543i)6-s + (0.418 + 0.241i)7-s + (0.851 − 0.524i)8-s + (−0.653 + 0.238i)9-s + (−0.545 − 1.45i)10-s + (0.837 − 0.483i)11-s + (0.482 + 0.266i)12-s + (0.309 − 1.75i)13-s + (−0.367 + 0.314i)14-s + (0.551 − 0.656i)15-s + (0.210 + 0.977i)16-s + (0.0983 + 0.0358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.300108 - 0.130509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.300108 - 0.130509i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 3.77i)T \) |
| 19 | \( 1 + (259. - 251. i)T \) |
good | 3 | \( 1 + (4.88 - 0.861i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (29.7 - 24.9i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-20.5 - 11.8i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-101. + 58.5i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-52.2 + 296. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-28.4 - 10.3i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-296. + 353. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (279. - 101. i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (602. + 348. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (278. + 1.57e3i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (914. + 1.09e3i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-1.06e3 - 2.93e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-728. - 611. i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-1.80e3 + 4.96e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (3.47e3 + 2.91e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (1.66e3 + 4.56e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-2.03e3 - 2.43e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-987. - 5.60e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-1.17e4 + 2.06e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (7.59e3 + 4.38e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (471. - 2.67e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (6.08e3 + 2.21e3i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−14.13280858104496034273560683360, −12.44117095240411270144624867865, −11.09477686559432675437431804311, −10.54670142890138413563988627531, −8.599830699644571521578672021417, −7.83050461006444670137854833474, −6.50843487308566601396606502986, −5.36458153513008641301531714343, −3.59922071652043619875787818834, −0.22529801545352043897730715876,
1.35942026606585078173846970203, 3.88985584927550550829114181984, 4.82992566558580629428121291714, 7.06423470849400616645788075479, 8.580696210767564487445121217634, 9.195031005424429508172876801732, 11.07476221943419337786694533045, 11.70642547695823113411584344167, 12.27487650314832825545968523948, 13.60708727318253194318961206035