L(s) = 1 | − 5-s − 2·7-s − 3·9-s − 11-s − 8·19-s − 8·23-s + 25-s + 10·29-s + 8·31-s + 2·35-s − 10·37-s − 2·41-s − 6·43-s + 3·45-s − 8·47-s − 3·49-s + 14·53-s + 55-s − 4·59-s + 10·61-s + 6·63-s + 4·67-s − 8·73-s + 2·77-s − 4·79-s + 9·81-s + 10·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.338·35-s − 1.64·37-s − 0.312·41-s − 0.914·43-s + 0.447·45-s − 1.16·47-s − 3/7·49-s + 1.92·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.755·63-s + 0.488·67-s − 0.936·73-s + 0.227·77-s − 0.450·79-s + 81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.52468211895507836187666788110, −9.991595366720817909982952019288, −8.479772891183006960706754270413, −8.327607038382254672069032807798, −6.74350248274942220173363652849, −6.11649847489499863225043030251, −4.79135750379205514131465277722, −3.59974368255131790788583382761, −2.43812931650948031191708956324, 0,
2.43812931650948031191708956324, 3.59974368255131790788583382761, 4.79135750379205514131465277722, 6.11649847489499863225043030251, 6.74350248274942220173363652849, 8.327607038382254672069032807798, 8.479772891183006960706754270413, 9.991595366720817909982952019288, 10.52468211895507836187666788110