L(s) = 1 | + (2.77 − 1.60i)3-s − 2.20i·7-s + (3.63 − 6.29i)9-s − 1.20·11-s + (−0.866 − 0.5i)13-s + (−1.85 + 1.07i)17-s + (−4.30 + 0.673i)19-s + (−3.53 − 6.11i)21-s + (−8.02 − 4.63i)23-s − 13.6i·27-s + (4.16 − 7.21i)29-s + 8.26·31-s + (−3.34 + 1.92i)33-s − 2.20i·37-s − 3.20·39-s + ⋯ |
L(s) = 1 | + (1.60 − 0.925i)3-s − 0.833i·7-s + (1.21 − 2.09i)9-s − 0.363·11-s + (−0.240 − 0.138i)13-s + (−0.449 + 0.259i)17-s + (−0.988 + 0.154i)19-s + (−0.770 − 1.33i)21-s + (−1.67 − 0.966i)23-s − 2.63i·27-s + (0.773 − 1.33i)29-s + 1.48·31-s + (−0.581 + 0.335i)33-s − 0.362i·37-s − 0.513·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.692386530\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.692386530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.30 - 0.673i)T \) |
good | 3 | \( 1 + (-2.77 + 1.60i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.85 - 1.07i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (8.02 + 4.63i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.16 + 7.21i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 2.20iT - 37T^{2} \) |
| 41 | \( 1 + (-3.30 - 5.72i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.03 - 1.17i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.21 - 4.16i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.232 - 0.134i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.16 + 7.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.70 + 2.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 1.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.23 - 9.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.20 + 2.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.20 - 14.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.73iT - 83T^{2} \) |
| 89 | \( 1 + (5.40 - 9.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.6 + 7.87i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.603141279854200837316573093790, −8.133828515807376099766920611091, −7.67871824797537442149239418808, −6.65751232934232647756798721867, −6.21051985465142360144767914331, −4.41499819923326684665332420642, −3.93952526303508092255428064917, −2.67852538842759472828819459813, −2.13458591754623908246234831743, −0.75848869116508844107064401442,
2.03091643410842390662532621536, 2.60780804010186684673832290669, 3.57583109611646783578014420473, 4.42350200874000904473138253266, 5.17433477450695156431219263839, 6.30060068604264536403664687525, 7.45818637068316299243435779503, 8.138576136781481362475073176785, 8.874482334755279521423186540867, 9.169940707133551118818303110173