| L(s) = 1 | + (−0.442 − 0.284i)2-s + (−0.784 − 5.45i)3-s + (−3.20 − 7.02i)4-s + (12.7 + 3.73i)5-s + (−1.20 + 2.63i)6-s + (−5.26 + 6.07i)7-s + (−1.17 + 8.18i)8-s + (−3.25 + 0.956i)9-s + (−4.57 − 5.27i)10-s + (−11.1 + 7.17i)11-s + (−35.8 + 23.0i)12-s + (47.1 + 54.3i)13-s + (4.05 − 1.19i)14-s + (10.4 − 72.4i)15-s + (−37.6 + 43.4i)16-s + (47.0 − 102. i)17-s + ⋯ |
| L(s) = 1 | + (−0.156 − 0.100i)2-s + (−0.151 − 1.05i)3-s + (−0.401 − 0.878i)4-s + (1.13 + 0.334i)5-s + (−0.0819 + 0.179i)6-s + (−0.284 + 0.328i)7-s + (−0.0520 + 0.361i)8-s + (−0.120 + 0.0354i)9-s + (−0.144 − 0.166i)10-s + (−0.306 + 0.196i)11-s + (−0.861 + 0.553i)12-s + (1.00 + 1.15i)13-s + (0.0774 − 0.0227i)14-s + (0.179 − 1.24i)15-s + (−0.587 + 0.678i)16-s + (0.670 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.861759 - 0.611898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.861759 - 0.611898i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (99.8 - 46.8i)T \) |
| good | 2 | \( 1 + (0.442 + 0.284i)T + (3.32 + 7.27i)T^{2} \) |
| 3 | \( 1 + (0.784 + 5.45i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-12.7 - 3.73i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (5.26 - 6.07i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (11.1 - 7.17i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-47.1 - 54.3i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-47.0 + 102. i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-8.05 - 17.6i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (32.0 - 70.1i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (34.1 - 237. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (251. - 73.7i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (110. + 32.4i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (50.2 + 349. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 385.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-182. + 210. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (329. + 380. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (91.6 - 637. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-397. - 255. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-141. - 91.1i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (277. + 606. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (541. + 624. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-789. + 231. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (80.1 + 557. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (20.4 + 6.01i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−17.71506844495973315162084645173, −15.91696789692392291869158814786, −14.03001851206585730328054075941, −13.64505666680785284534348579926, −11.99902032624380335510574252927, −10.27176452224165623654621172083, −9.114484211257577419827325804122, −6.84416032453293300935723270283, −5.63798327709881139582111950364, −1.71794375015945904026701848559,
3.82362366911442314618669545760, 5.73491591967016608016129413555, 8.183149629114134586733071988309, 9.628588713661632982955420785224, 10.58201142611090007510790609828, 12.75314748148900687949307426506, 13.59872380343852424139459024902, 15.42975610455356190848859428685, 16.58029769108293020389017835640, 17.31499973614596008396327154400
