L(s) = 1 | − 2i·2-s + i·3-s − 2·4-s + (−2 + i)5-s + 2·6-s + i·7-s + 2·9-s + (2 + 4i)10-s − 3·11-s − 2i·12-s + i·13-s + 2·14-s + (−1 − 2i)15-s − 4·16-s − 7i·17-s − 4i·18-s + ⋯ |
L(s) = 1 | − 1.41i·2-s + 0.577i·3-s − 4-s + (−0.894 + 0.447i)5-s + 0.816·6-s + 0.377i·7-s + 0.666·9-s + (0.632 + 1.26i)10-s − 0.904·11-s − 0.577i·12-s + 0.277i·13-s + 0.534·14-s + (−0.258 − 0.516i)15-s − 16-s − 1.69i·17-s − 0.942i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.581647 - 0.359478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.581647 - 0.359478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−15.98210910707013386448206162217, −15.51459514519935734353166136015, −13.73908249804327735744761172916, −12.36148837795374284937339794164, −11.42621865015752539898361587727, −10.42068121271810089671381063107, −9.316604718406395525588995501970, −7.35689965275207777678389254869, −4.60140633504418596338695192636, −3.02592800233480114290664971806,
4.58669105072248681251752024317, 6.41057499424661595037771355877, 7.69057201102210773213766625761, 8.382229318624263597580461235223, 10.57199601674620508871050040716, 12.39216276691738128396285050555, 13.34283107156104634236933164891, 14.84551408076473918952506948409, 15.70323106477834300119154299758, 16.58448451363154192517734140835