L(s) = 1 | + 1.51·5-s + i·7-s + 0.935i·11-s − 2.14i·13-s − 2.62i·17-s − 3.46·19-s − 6.86·23-s − 2.70·25-s − 2.44·29-s − 2i·31-s + 1.51i·35-s + 2.14i·37-s − 2.62i·41-s − 7.74·43-s − 49-s + ⋯ |
L(s) = 1 | + 0.677·5-s + 0.377i·7-s + 0.282i·11-s − 0.593i·13-s − 0.635i·17-s − 0.794·19-s − 1.43·23-s − 0.541·25-s − 0.454·29-s − 0.359i·31-s + 0.255i·35-s + 0.351i·37-s − 0.409i·41-s − 1.18·43-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3081428850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3081428850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 - 0.935iT - 11T^{2} \) |
| 13 | \( 1 + 2.14iT - 13T^{2} \) |
| 17 | \( 1 + 2.62iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 2.14iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.79iT - 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 - 1.87iT - 83T^{2} \) |
| 89 | \( 1 - 2.62iT - 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.122421925571395592064199102464, −7.46965701348378370883447513507, −6.47569870496926241787645912116, −5.93940145667382458559464509505, −5.21776115117800848374165057503, −4.37473555531120676799514180839, −3.40073265631266067364896305148, −2.38632120213812312037036955839, −1.69564848554182173183661525418, −0.07813937921574475662272113579,
1.55868587331345021541396932010, 2.20643458358369462800550282282, 3.46615584439160411129009408092, 4.17211360940471675651001094743, 5.02722138048695034396015899442, 6.08887426509650168653110305036, 6.29851293940383266326176738634, 7.31605858086132148292585101503, 8.095275385399348754986973085730, 8.721918727414541249650430627255