| L(s) = 1 | + (−4.76 + 1.39i)2-s + (−0.541 − 0.625i)3-s + (14.0 − 9.02i)4-s + (−2.93 − 20.3i)5-s + (3.45 + 2.22i)6-s + (−4.70 + 10.3i)7-s + (−28.2 + 32.6i)8-s + (3.74 − 26.0i)9-s + (42.5 + 93.0i)10-s + (−29.5 − 8.68i)11-s + (−13.2 − 3.89i)12-s + (−5.14 − 11.2i)13-s + (8.01 − 55.7i)14-s + (−11.1 + 12.8i)15-s + (33.6 − 73.7i)16-s + (28.3 + 18.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.68 + 0.494i)2-s + (−0.104 − 0.120i)3-s + (1.75 − 1.12i)4-s + (−0.262 − 1.82i)5-s + (0.235 + 0.151i)6-s + (−0.254 + 0.556i)7-s + (−1.24 + 1.44i)8-s + (0.138 − 0.964i)9-s + (1.34 + 2.94i)10-s + (−0.810 − 0.238i)11-s + (−0.318 − 0.0935i)12-s + (−0.109 − 0.240i)13-s + (0.152 − 1.06i)14-s + (−0.192 + 0.221i)15-s + (0.526 − 1.15i)16-s + (0.404 + 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.313085 - 0.277205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313085 - 0.277205i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-31.2 - 105. i)T \) |
| good | 2 | \( 1 + (4.76 - 1.39i)T + (6.73 - 4.32i)T^{2} \) |
| 3 | \( 1 + (0.541 + 0.625i)T + (-3.84 + 26.7i)T^{2} \) |
| 5 | \( 1 + (2.93 + 20.3i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (4.70 - 10.3i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (29.5 + 8.68i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (5.14 + 11.2i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-28.3 - 18.2i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-85.9 + 55.2i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-57.9 - 37.2i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-120. + 139. i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-42.8 + 297. i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-9.50 - 66.0i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-66.3 - 76.5i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 72.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-125. + 274. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (-27.6 - 60.6i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-326. + 376. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (421. - 123. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-479. + 140. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (491. - 315. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (12.4 + 27.3i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-38.2 + 266. i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (238. + 275. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (65.9 + 458. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−17.20721529636486825293900221117, −16.04436585089015715288732493162, −15.49028316159331044334267254362, −12.91966210024159603632493324174, −11.68447931449390150218920006473, −9.671431340782122253544153652467, −8.878373425437757381907351106788, −7.68924500037341922610670670734, −5.63901845107255374034732742721, −0.77336363075243149060801338309,
2.77280308192198055845967132588, 7.00056191577447123334815610578, 7.891195188695209043198806163154, 10.16944023275910034986276712746, 10.46308953933247232410305094352, 11.71240530543523504435922864752, 13.97414093179804272484971750664, 15.63317781622655632697008417073, 16.70555145847933110485581776606, 18.09295058210345810394519845315
