| L(s) = 1 | + (0.123 + 0.861i)2-s + (−1.04 + 2.28i)3-s + (3.11 − 0.913i)4-s + (−1.80 − 1.56i)5-s + (−2.09 − 0.615i)6-s + (−6.61 − 10.2i)7-s + (2.61 + 5.73i)8-s + (1.77 + 2.04i)9-s + (1.12 − 1.75i)10-s + (−4.49 − 0.646i)11-s + (−1.15 + 8.05i)12-s + (1.78 + 1.14i)13-s + (8.04 − 6.97i)14-s + (5.46 − 2.49i)15-s + (6.30 − 4.04i)16-s + (−6.95 + 23.6i)17-s + ⋯ |
| L(s) = 1 | + (0.0619 + 0.430i)2-s + (−0.347 + 0.760i)3-s + (0.777 − 0.228i)4-s + (−0.361 − 0.313i)5-s + (−0.349 − 0.102i)6-s + (−0.945 − 1.47i)7-s + (0.327 + 0.716i)8-s + (0.196 + 0.227i)9-s + (0.112 − 0.175i)10-s + (−0.408 − 0.0587i)11-s + (−0.0964 + 0.671i)12-s + (0.137 + 0.0882i)13-s + (0.574 − 0.498i)14-s + (0.364 − 0.166i)15-s + (0.393 − 0.253i)16-s + (−0.408 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.846151 + 0.310191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846151 + 0.310191i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (14.6 + 17.7i)T \) |
| good | 2 | \( 1 + (-0.123 - 0.861i)T + (-3.83 + 1.12i)T^{2} \) |
| 3 | \( 1 + (1.04 - 2.28i)T + (-5.89 - 6.80i)T^{2} \) |
| 5 | \( 1 + (1.80 + 1.56i)T + (3.55 + 24.7i)T^{2} \) |
| 7 | \( 1 + (6.61 + 10.2i)T + (-20.3 + 44.5i)T^{2} \) |
| 11 | \( 1 + (4.49 + 0.646i)T + (116. + 34.0i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 1.14i)T + (70.2 + 153. i)T^{2} \) |
| 17 | \( 1 + (6.95 - 23.6i)T + (-243. - 156. i)T^{2} \) |
| 19 | \( 1 + (-3.08 - 10.5i)T + (-303. + 195. i)T^{2} \) |
| 29 | \( 1 + (-29.6 - 8.69i)T + (707. + 454. i)T^{2} \) |
| 31 | \( 1 + (0.235 + 0.514i)T + (-629. + 726. i)T^{2} \) |
| 37 | \( 1 + (-10.3 + 8.98i)T + (194. - 1.35e3i)T^{2} \) |
| 41 | \( 1 + (-38.9 + 44.9i)T + (-239. - 1.66e3i)T^{2} \) |
| 43 | \( 1 + (-67.7 - 30.9i)T + (1.21e3 + 1.39e3i)T^{2} \) |
| 47 | \( 1 + 44.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (11.9 + 18.5i)T + (-1.16e3 + 2.55e3i)T^{2} \) |
| 59 | \( 1 + (57.2 + 36.7i)T + (1.44e3 + 3.16e3i)T^{2} \) |
| 61 | \( 1 + (17.5 - 8.01i)T + (2.43e3 - 2.81e3i)T^{2} \) |
| 67 | \( 1 + (-50.4 + 7.25i)T + (4.30e3 - 1.26e3i)T^{2} \) |
| 71 | \( 1 + (4.48 + 31.1i)T + (-4.83e3 + 1.42e3i)T^{2} \) |
| 73 | \( 1 + (77.3 - 22.7i)T + (4.48e3 - 2.88e3i)T^{2} \) |
| 79 | \( 1 + (57.8 - 90.0i)T + (-2.59e3 - 5.67e3i)T^{2} \) |
| 83 | \( 1 + (-75.8 + 65.7i)T + (980. - 6.81e3i)T^{2} \) |
| 89 | \( 1 + (61.6 + 28.1i)T + (5.18e3 + 5.98e3i)T^{2} \) |
| 97 | \( 1 + (-44.7 - 38.7i)T + (1.33e3 + 9.31e3i)T^{2} \) |
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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
−17.22296814120240895644422255813, −16.20370812018694005122421434925, −15.88452246757529689522146707410, −14.23722844294057224432828280285, −12.69515059500453080490775285114, −10.80741864171200215860091721992, −10.19925518964009573980620776124, −7.80976414183668124697155320995, −6.28797425976800747300867470837, −4.21601322484043912352973613980,
2.80083536087224165167132013008, 6.14949212947949305553521821003, 7.38390044104242064944246867375, 9.523124680859139248009251612486, 11.42705832771448754803413402253, 12.18464129221690353252393087676, 13.18667252101633743251174990129, 15.46659154920637312177774215326, 15.94223102322728513083898380953, 17.89870680429327636864888468007
