L(s) = 1 | − 3·5-s + 4·11-s + 13-s + 17-s + 4·23-s + 4·25-s − 29-s + 3·31-s + 2·37-s + 5·41-s − 6·43-s + 9·47-s − 2·53-s − 12·55-s + 59-s + 12·61-s − 3·65-s − 10·67-s − 8·73-s − 5·83-s − 3·85-s + 6·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 1.20·11-s + 0.277·13-s + 0.242·17-s + 0.834·23-s + 4/5·25-s − 0.185·29-s + 0.538·31-s + 0.328·37-s + 0.780·41-s − 0.914·43-s + 1.31·47-s − 0.274·53-s − 1.61·55-s + 0.130·59-s + 1.53·61-s − 0.372·65-s − 1.22·67-s − 0.936·73-s − 0.548·83-s − 0.325·85-s + 0.635·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−13.83105157847444, −13.31849571499886, −12.71568017319733, −12.28821707416686, −11.83455598370900, −11.34330236233144, −11.20801865651829, −10.42726921903755, −9.965755601261248, −9.276520844886391, −8.836207967047126, −8.482494940325552, −7.771328985556078, −7.478253178101336, −6.837102971333142, −6.472192035494777, −5.769347979140913, −5.190365075565288, −4.377862785440428, −4.170659552380281, −3.560399016706408, −3.066235762416317, −2.338862061336828, −1.344711023045666, −0.9068288844910222, 0,
0.9068288844910222, 1.344711023045666, 2.338862061336828, 3.066235762416317, 3.560399016706408, 4.170659552380281, 4.377862785440428, 5.190365075565288, 5.769347979140913, 6.472192035494777, 6.837102971333142, 7.478253178101336, 7.771328985556078, 8.482494940325552, 8.836207967047126, 9.276520844886391, 9.965755601261248, 10.42726921903755, 11.20801865651829, 11.34330236233144, 11.83455598370900, 12.28821707416686, 12.71568017319733, 13.31849571499886, 13.83105157847444