L(s) = 1 | − 1.53i·2-s − 1.17i·3-s − 0.369·4-s − 1.80·6-s − i·7-s − 2.51i·8-s + 1.63·9-s − 11-s + 0.431i·12-s + 0.0917i·13-s − 1.53·14-s − 4.60·16-s − 5.51i·17-s − 2.51i·18-s − 0.921·19-s + ⋯ |
L(s) = 1 | − 1.08i·2-s − 0.675i·3-s − 0.184·4-s − 0.735·6-s − 0.377i·7-s − 0.887i·8-s + 0.543·9-s − 0.301·11-s + 0.124i·12-s + 0.0254i·13-s − 0.411·14-s − 1.15·16-s − 1.33i·17-s − 0.591i·18-s − 0.211·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.656171242\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.656171242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.53iT - 2T^{2} \) |
| 3 | \( 1 + 1.17iT - 3T^{2} \) |
| 13 | \( 1 - 0.0917iT - 13T^{2} \) |
| 17 | \( 1 + 5.51iT - 17T^{2} \) |
| 19 | \( 1 + 0.921T + 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 0.879T + 31T^{2} \) |
| 37 | \( 1 + 8.78iT - 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 + 3.86iT - 43T^{2} \) |
| 47 | \( 1 + 5.90iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.52iT - 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−9.055152432507200663070970832142, −7.71335422400674655626875964579, −7.28210593931392936959772763112, −6.58580302306909642864462596412, −5.46261222213055677785329790620, −4.37227504869229364848533794137, −3.51896121020202418661748651740, −2.50615228947633431354749355612, −1.64622089465333369012867550172, −0.58261001481125712865428619495,
1.76918009427112357994026879513, 2.97419947414149516084962456817, 4.21584355589939830284473724345, 4.87818687222806296904423685216, 5.78261168963361206416236811009, 6.47351840455183722177426339970, 7.18044535138138921866447342136, 8.254372940546702779648823266293, 8.488242707522817088226436524449, 9.590012460223596045597758548553