L(s) = 1 | + 3-s + 9-s + 2·11-s + 4·13-s + 27-s − 2·29-s + 10·31-s + 2·33-s + 4·37-s + 4·39-s + 6·41-s − 4·43-s − 7·49-s + 2·53-s + 6·59-s − 6·61-s − 4·67-s + 8·71-s − 14·73-s + 2·79-s + 81-s − 4·83-s − 2·87-s + 14·89-s + 10·93-s + 6·97-s + 2·99-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.348·33-s + 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s + 0.781·59-s − 0.768·61-s − 0.488·67-s + 0.949·71-s − 1.63·73-s + 0.225·79-s + 1/9·81-s − 0.439·83-s − 0.214·87-s + 1.48·89-s + 1.03·93-s + 0.609·97-s + 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.128377112\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.128377112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−7.895807958329825858083533938517, −6.96040749586959779936163216505, −6.36207060538630857186029968352, −5.79734763457005003596102756702, −4.75165984982239360455620772060, −4.13703167656453940401462995855, −3.41917487768383339612667258558, −2.69441255516611215843698956517, −1.69506967887872650017562137107, −0.868412319694524296811567863797,
0.868412319694524296811567863797, 1.69506967887872650017562137107, 2.69441255516611215843698956517, 3.41917487768383339612667258558, 4.13703167656453940401462995855, 4.75165984982239360455620772060, 5.79734763457005003596102756702, 6.36207060538630857186029968352, 6.96040749586959779936163216505, 7.895807958329825858083533938517