L(s) = 1 | + (−0.866 − 0.5i)3-s + (−2.29 + 1.32i)7-s + (0.499 + 0.866i)9-s + (0.822 − 1.42i)11-s − 2.64i·13-s + (−1.42 − 0.822i)17-s + (4.14 + 7.18i)19-s + 2.64·21-s + (1.42 − 0.822i)23-s − 0.999i·27-s − 7.64·29-s + (2.14 − 3.71i)31-s + (−1.42 + 0.822i)33-s + (−0.559 + 0.322i)37-s + (−1.32 + 2.29i)39-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (−0.866 + 0.499i)7-s + (0.166 + 0.288i)9-s + (0.248 − 0.429i)11-s − 0.733i·13-s + (−0.345 − 0.199i)17-s + (0.951 + 1.64i)19-s + 0.577·21-s + (0.297 − 0.171i)23-s − 0.192i·27-s − 1.41·29-s + (0.385 − 0.667i)31-s + (−0.248 + 0.143i)33-s + (−0.0919 + 0.0530i)37-s + (−0.211 + 0.366i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8770419567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8770419567\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.29 - 1.32i)T \) |
good | 11 | \( 1 + (-0.822 + 1.42i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.64iT - 13T^{2} \) |
| 17 | \( 1 + (1.42 + 0.822i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.14 - 7.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.42 + 0.822i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.559 - 0.322i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 + 5.93iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.85 + 1.64i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.46 + 9.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.559 - 0.322i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + (11.4 + 6.61i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.14 - 1.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.9iT - 83T^{2} \) |
| 89 | \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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
−8.958699561803741134240566015115, −8.010178991920593573241107057699, −7.36437924966998761787848790491, −6.33541652410325893913534653468, −5.84225920419664343222361752427, −5.12964194614714988610414921105, −3.80566009899217009417076343302, −3.07603451184552925529554082822, −1.81631873583150509647258055526, −0.37954326800083424113225812708,
1.09935860618127219007856242167, 2.60043835360970783003067927992, 3.65696374562989693414014619802, 4.45697065034218007780297751342, 5.26803665095405792400545040459, 6.28645767568046472676021102219, 6.95394346898707860958609958258, 7.45884982632252398829251811073, 8.820575819529908908249811745950, 9.421479447285227764033232896531