L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 2·11-s + 13-s + 2·15-s + 2·17-s − 21-s − 25-s − 27-s − 2·29-s + 2·33-s − 2·35-s + 37-s − 39-s + 43-s − 2·45-s + 4·47-s − 6·49-s − 2·51-s + 6·53-s + 4·55-s + 8·59-s + 61-s + 63-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.348·33-s − 0.338·35-s + 0.164·37-s − 0.160·39-s + 0.152·43-s − 0.298·45-s + 0.583·47-s − 6/7·49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.04·59-s + 0.128·61-s + 0.125·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.42994405447312, −16.09124013161308, −15.50763118560653, −14.96604888798346, −14.48194442781752, −13.62297153118489, −13.18313135849600, −12.43702749440930, −12.00438460017363, −11.38183581499391, −11.03566743007182, −10.29683587809196, −9.824869239328240, −8.902609561274123, −8.339154763367204, −7.621087823946684, −7.352148113091306, −6.440773120963800, −5.725844157769832, −5.169827542517263, −4.402911119017581, −3.824277992594454, −3.023711053660488, −2.020823765864921, −0.9944734700913198, 0,
0.9944734700913198, 2.020823765864921, 3.023711053660488, 3.824277992594454, 4.402911119017581, 5.169827542517263, 5.725844157769832, 6.440773120963800, 7.352148113091306, 7.621087823946684, 8.339154763367204, 8.902609561274123, 9.824869239328240, 10.29683587809196, 11.03566743007182, 11.38183581499391, 12.00438460017363, 12.43702749440930, 13.18313135849600, 13.62297153118489, 14.48194442781752, 14.96604888798346, 15.50763118560653, 16.09124013161308, 16.42994405447312