L(s) = 1 | − 0.471i·2-s − 1.04i·3-s + 1.77·4-s + (0.713 + 2.11i)5-s − 0.490·6-s − 1.17i·7-s − 1.78i·8-s + 1.91·9-s + (1 − 0.336i)10-s + 0.713·11-s − 1.84i·12-s − 4.10i·13-s − 0.554·14-s + (2.20 − 0.742i)15-s + 2.71·16-s − 2.55i·17-s + ⋯ |
L(s) = 1 | − 0.333i·2-s − 0.600i·3-s + 0.888·4-s + (0.319 + 0.947i)5-s − 0.200·6-s − 0.444i·7-s − 0.630i·8-s + 0.639·9-s + (0.316 − 0.106i)10-s + 0.215·11-s − 0.533i·12-s − 1.13i·13-s − 0.148·14-s + (0.569 − 0.191i)15-s + 0.678·16-s − 0.619i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.525697406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.525697406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.713 - 2.11i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.471iT - 2T^{2} \) |
| 3 | \( 1 + 1.04iT - 3T^{2} \) |
| 7 | \( 1 + 1.17iT - 7T^{2} \) |
| 11 | \( 1 - 0.713T + 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 + 2.55iT - 17T^{2} \) |
| 23 | \( 1 + 0.607iT - 23T^{2} \) |
| 29 | \( 1 + 0.859T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 9.38iT - 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 4.98iT - 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 5.18iT - 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 16.7iT - 97T^{2} \) |
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
−9.373821362208675257683492471277, −7.84577495313600803435078418314, −7.58008992800278923768640910874, −6.63909820795991376605061361042, −6.33058098873188813840905757425, −5.14476769482400780947940293283, −3.76518645573577522972737080865, −2.96253167396537815354826321319, −2.08070592906829012526750993060, −0.993805234174546390443393751249,
1.47619501254604508180548067576, 2.27435741086681025891838689856, 3.77937264845429953006652327702, 4.50661818192185881887266643711, 5.50818366053541696473848581051, 6.09869500084593781361510068379, 7.07192806452476446386886758576, 7.76983382002409522749370955553, 8.932644837663469870730314790734, 9.169290931200554888489299962367