L(s) = 1 | + (0.707 − 0.707i)3-s + (2.63 − 0.235i)7-s − 1.00i·9-s + 4.99·11-s + (−2.63 + 2.63i)13-s + (4.14 + 4.14i)17-s + 4.66·19-s + (1.69 − 2.03i)21-s + (−4.41 − 4.41i)23-s + (−0.707 − 0.707i)27-s + 2.34i·29-s − 2.57i·31-s + (3.53 − 3.53i)33-s + (−7.06 + 7.06i)37-s + 3.72i·39-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.996 − 0.0891i)7-s − 0.333i·9-s + 1.50·11-s + (−0.729 + 0.729i)13-s + (1.00 + 1.00i)17-s + 1.06·19-s + (0.370 − 0.443i)21-s + (−0.921 − 0.921i)23-s + (−0.136 − 0.136i)27-s + 0.436i·29-s − 0.461i·31-s + (0.614 − 0.614i)33-s + (−1.16 + 1.16i)37-s + 0.595i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.569088622\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.569088622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.235i)T \) |
good | 11 | \( 1 - 4.99T + 11T^{2} \) |
| 13 | \( 1 + (2.63 - 2.63i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.14 - 4.14i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 + (4.41 + 4.41i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.34iT - 29T^{2} \) |
| 31 | \( 1 + 2.57iT - 31T^{2} \) |
| 37 | \( 1 + (7.06 - 7.06i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.87iT - 41T^{2} \) |
| 43 | \( 1 + (-2.59 - 2.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.813 - 0.813i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.0317 - 0.0317i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.858T + 59T^{2} \) |
| 61 | \( 1 + 1.59iT - 61T^{2} \) |
| 67 | \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + (-7.92 + 7.92i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.69iT - 79T^{2} \) |
| 83 | \( 1 + (-5.50 + 5.50i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 + (-3.51 - 3.51i)T + 97iT^{2} \) |
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
−9.052359236845721723920882924361, −8.194278130257423240133082345528, −7.68457830594052476491411231781, −6.76593525625119648785362461824, −6.10458498941487585073143175890, −4.96743687522464483650388798677, −4.16505036352849628579365812359, −3.28208575370054368936670620952, −1.92446362578728756608507294630, −1.24211734423064442324133892019,
1.08047989678845290713978186460, 2.24121471266022334973897182486, 3.42021509072612403276674293117, 4.11041684349550437686752575214, 5.26626784455504005227600705459, 5.59266546438751170486047409153, 7.11163207792730159875474590607, 7.52527564871503839434213426834, 8.389767813819083665651952300494, 9.200694348862473874807865164976