L(s) = 1 | + 3·5-s + 7-s − 2·11-s − 4·13-s − 5·17-s + 4·23-s + 4·25-s − 4·29-s − 8·31-s + 3·35-s + 37-s − 7·41-s + 5·43-s − 47-s + 49-s − 2·53-s − 6·55-s + 11·59-s − 14·61-s − 12·65-s − 4·67-s − 12·71-s − 2·77-s + 13·79-s − 11·83-s − 15·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s − 1.21·17-s + 0.834·23-s + 4/5·25-s − 0.742·29-s − 1.43·31-s + 0.507·35-s + 0.164·37-s − 1.09·41-s + 0.762·43-s − 0.145·47-s + 1/7·49-s − 0.274·53-s − 0.809·55-s + 1.43·59-s − 1.79·61-s − 1.48·65-s − 0.488·67-s − 1.42·71-s − 0.227·77-s + 1.46·79-s − 1.20·83-s − 1.62·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−7.55914698657869452684205479125, −7.06753570730490724834367427121, −6.25294027428831838563205442446, −5.44960671348277267349079920488, −5.04813970842606167091023185214, −4.19763628740708362161192488205, −2.95775652327444994986043768987, −2.24170867221360265787967931889, −1.59580518356912212039355325381, 0,
1.59580518356912212039355325381, 2.24170867221360265787967931889, 2.95775652327444994986043768987, 4.19763628740708362161192488205, 5.04813970842606167091023185214, 5.44960671348277267349079920488, 6.25294027428831838563205442446, 7.06753570730490724834367427121, 7.55914698657869452684205479125