L(s) = 1 | − 2·7-s − 3·9-s + 6·11-s − 2·13-s + 6·17-s − 2·19-s − 6·23-s + 6·29-s + 4·31-s − 6·37-s − 2·41-s − 4·43-s + 10·47-s − 3·49-s − 2·53-s + 10·59-s − 10·61-s + 6·63-s + 4·67-s + 16·71-s + 6·73-s − 12·77-s + 9·81-s + 8·83-s + 6·89-s + 4·91-s − 2·97-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 9-s + 1.80·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 1.11·29-s + 0.718·31-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.45·47-s − 3/7·49-s − 0.274·53-s + 1.30·59-s − 1.28·61-s + 0.755·63-s + 0.488·67-s + 1.89·71-s + 0.702·73-s − 1.36·77-s + 81-s + 0.878·83-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.590422329\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590422329\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−8.619813424147769520748181429566, −8.084965199851113522138028897679, −7.01584367895983813721050146736, −6.36430682298437739959297067305, −5.83010083830286361833053087012, −4.82009481951792433465324245970, −3.76905487274006000859901008652, −3.22223130405946699984035495822, −2.08126057870373217260484368812, −0.75557259351887432839390395284,
0.75557259351887432839390395284, 2.08126057870373217260484368812, 3.22223130405946699984035495822, 3.76905487274006000859901008652, 4.82009481951792433465324245970, 5.83010083830286361833053087012, 6.36430682298437739959297067305, 7.01584367895983813721050146736, 8.084965199851113522138028897679, 8.619813424147769520748181429566