L(s) = 1 | + 3-s − 3·7-s + 9-s + 3·11-s + 13-s + 7·17-s + 8·19-s − 3·21-s − 4·23-s + 27-s − 3·29-s + 11·31-s + 3·33-s + 39-s − 2·41-s + 8·43-s + 9·47-s + 2·49-s + 7·51-s + 9·53-s + 8·57-s + 9·59-s + 61-s − 3·63-s − 5·67-s − 4·69-s − 12·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s + 1.83·19-s − 0.654·21-s − 0.834·23-s + 0.192·27-s − 0.557·29-s + 1.97·31-s + 0.522·33-s + 0.160·39-s − 0.312·41-s + 1.21·43-s + 1.31·47-s + 2/7·49-s + 0.980·51-s + 1.23·53-s + 1.05·57-s + 1.17·59-s + 0.128·61-s − 0.377·63-s − 0.610·67-s − 0.481·69-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.606907016\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.606907016\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−15.04624301628963, −14.46262052956719, −14.00434336760336, −13.58596327998439, −13.14669633138554, −12.22160074241520, −12.04148857952965, −11.65093985932066, −10.58893946137829, −10.09321216199351, −9.662131509087371, −9.290605819576386, −8.650061887817196, −7.911759432257784, −7.471749786926114, −6.902438938014281, −6.151361155052708, −5.760060109128253, −5.032683085916360, −3.987806672343283, −3.685167196453286, −3.037494068310573, −2.457991712523340, −1.268712457382853, −0.8079693696848023,
0.8079693696848023, 1.268712457382853, 2.457991712523340, 3.037494068310573, 3.685167196453286, 3.987806672343283, 5.032683085916360, 5.760060109128253, 6.151361155052708, 6.902438938014281, 7.471749786926114, 7.911759432257784, 8.650061887817196, 9.290605819576386, 9.662131509087371, 10.09321216199351, 10.58893946137829, 11.65093985932066, 12.04148857952965, 12.22160074241520, 13.14669633138554, 13.58596327998439, 14.00434336760336, 14.46262052956719, 15.04624301628963