| L(s) = 1 | − 2i·3-s + 4i·5-s + 7-s − 9-s + 8·15-s − 2·17-s + 2i·19-s − 2i·21-s + 8·23-s − 11·25-s − 4i·27-s + 2i·29-s − 4·31-s + 4i·35-s + 6i·37-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 1.78i·5-s + 0.377·7-s − 0.333·9-s + 2.06·15-s − 0.485·17-s + 0.458i·19-s − 0.436i·21-s + 1.66·23-s − 2.20·25-s − 0.769i·27-s + 0.371i·29-s − 0.718·31-s + 0.676i·35-s + 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.686606098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.686606098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 - 4iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.388765843764168056009404609879, −8.338230514783433788065511216967, −7.48936401228592929393332306087, −7.06068560595911632697929483495, −6.47379165362217690725562152793, −5.65193114840502434361126362418, −4.32376748799184881771168276123, −3.12781498027044281335497555907, −2.43757830332867659119139376806, −1.32957081894989806118358517983,
0.67948795505594276276851091436, 2.00755734194792644921126432671, 3.60743973957802595221786926123, 4.34682324951894699210980527070, 5.09767616527127482135480963821, 5.38388055628819486694203169593, 6.81117659057003561668218897538, 7.85892418340992681283929690898, 8.710065898488266832508227332926, 9.203039019592807011678450681956
