L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.62 + 0.358i)7-s − 0.999i·8-s + (−2.59 + 1.5i)11-s + 2.44i·13-s + (2.44 + i)14-s + (−0.5 + 0.866i)16-s + (2.95 + 5.12i)17-s + (−5.12 − 2.95i)19-s + 3·22-s + (−3.67 − 2.12i)23-s + (1.22 − 2.12i)26-s + (−1.62 − 2.09i)28-s − 7.24i·29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.990 + 0.135i)7-s − 0.353i·8-s + (−0.783 + 0.452i)11-s + 0.679i·13-s + (0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.717 + 1.24i)17-s + (−1.17 − 0.678i)19-s + 0.639·22-s + (−0.766 − 0.442i)23-s + (0.240 − 0.416i)26-s + (−0.306 − 0.395i)28-s − 1.34i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5787764919\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5787764919\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.358i)T \) |
good | 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (-2.95 - 5.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.242T + 43T^{2} \) |
| 47 | \( 1 + (-2.95 + 5.12i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.27 + 3.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.03 - 6.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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
−8.684092342216333875566000649297, −7.82283273346788174181445636803, −7.12376792539545159599590697112, −6.29918796144538775317491725389, −5.67859243494536862018872342024, −4.36059444641613049764801519917, −3.73367452017986834486882804579, −2.58954535698109018311234781500, −1.92325686253028341240956056872, −0.29574091581702251528318757179,
0.792971381362917272931947788805, 2.31668643047815727870522719184, 3.16898121966913030585047533929, 4.09586541496072180862625666408, 5.47094845609024238215398754536, 5.74521764255956550861677182234, 6.70775477061188036023846313028, 7.54228171024679255469400081901, 7.934381338950786172514166853342, 8.963999523610793005419934787688