| L(s) = 1 | + 3.23i·5-s − 4.57i·7-s + 1.23i·11-s + (2.23 + 2.82i)13-s + 5.65·17-s − 1.08i·19-s − 3.49·23-s − 5.47·25-s + 9.15·29-s + 4.57i·31-s + 14.8·35-s + 5.65i·37-s + 5.70i·41-s + 8.94·43-s − 2.76i·47-s + ⋯ |
| L(s) = 1 | + 1.44i·5-s − 1.72i·7-s + 0.372i·11-s + (0.620 + 0.784i)13-s + 1.37·17-s − 0.247i·19-s − 0.728·23-s − 1.09·25-s + 1.69·29-s + 0.821i·31-s + 2.50·35-s + 0.929i·37-s + 0.891i·41-s + 1.36·43-s − 0.403i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55344 + 0.539886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55344 + 0.539886i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-2.23 - 2.82i)T \) |
| good | 5 | \( 1 - 3.23iT - 5T^{2} \) |
| 7 | \( 1 + 4.57iT - 7T^{2} \) |
| 11 | \( 1 - 1.23iT - 11T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 1.08iT - 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 - 4.57iT - 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 - 5.70iT - 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 2.76iT - 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 - 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 7.70iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 6.76iT - 83T^{2} \) |
| 89 | \( 1 + 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 6.99iT - 97T^{2} \) |
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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
−10.29146679107631334803581120115, −9.673534082361530862068115976829, −8.242713889446877708788604138501, −7.43470213770697361647838329808, −6.82521877483428054734230200470, −6.17607456191662042303436374691, −4.62959445026710170802449249961, −3.75652483365564900053142638650, −2.90804947809190257184968375972, −1.26830161863694313553830468636,
0.947411682839192209144226630666, 2.35910521890706450960969423766, 3.60464027420102415877274438063, 4.90965363854962623864501342258, 5.65249742644982580696421623966, 6.09194593704837248672289939000, 7.910187274099124836331081046680, 8.307707853621333711375548374715, 9.070069285769126268577974784475, 9.703481506897249268413914115433
