L(s) = 1 | − 0.646i·3-s + 0.646·5-s + (−2.44 − i)7-s + 2.58·9-s − 3.58·11-s + 5.54·13-s − 0.417i·15-s + 6.19i·17-s + 6.83i·19-s + (−0.646 + 1.58i)21-s + 5.58i·23-s − 4.58·25-s − 3.60i·27-s − 6i·29-s + 6.19·31-s + ⋯ |
L(s) = 1 | − 0.373i·3-s + 0.288·5-s + (−0.925 − 0.377i)7-s + 0.860·9-s − 1.08·11-s + 1.53·13-s − 0.107i·15-s + 1.50i·17-s + 1.56i·19-s + (−0.140 + 0.345i)21-s + 1.16i·23-s − 0.916·25-s − 0.694i·27-s − 1.11i·29-s + 1.11·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.652316345\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.652316345\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
good | 3 | \( 1 + 0.646iT - 3T^{2} \) |
| 5 | \( 1 - 0.646T + 5T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 - 6.19iT - 17T^{2} \) |
| 19 | \( 1 - 6.83iT - 19T^{2} \) |
| 23 | \( 1 - 5.58iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 1.29iT - 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + 9.16iT - 53T^{2} \) |
| 59 | \( 1 + 5.54iT - 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.41T + 67T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 - 7.48iT - 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + 0.646iT - 83T^{2} \) |
| 89 | \( 1 - 7.48iT - 89T^{2} \) |
| 97 | \( 1 - 18.5iT - 97T^{2} \) |
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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
−9.558469152622834421854669047210, −8.157673873511094785446685913232, −8.016933819447705373824596513533, −6.85237744510143742027134151845, −6.08463089524824816643026133262, −5.62823070904916770844638990069, −3.98789237878459229857784592053, −3.67710925102367295729955137250, −2.18112729203969994249443034554, −1.14808382048185877774186369158,
0.72006819822398289212846816617, 2.43271635073889260157343784549, 3.16774087700123960125128754572, 4.32321717842624368711766647987, 5.11583770347137561438641497540, 6.03825032581550080130113311180, 6.81739729506876811239593505815, 7.54148964757729746540440413241, 8.757294595733494980737727013895, 9.160367073053877763312478555755