L(s) = 1 | − 2-s − 2·3-s − 4-s − 3·5-s + 2·6-s + 7-s + 3·8-s + 9-s + 3·10-s − 5·11-s + 2·12-s + 13-s − 14-s + 6·15-s − 16-s + 4·17-s − 18-s − 4·19-s + 3·20-s − 2·21-s + 5·22-s − 9·23-s − 6·24-s + 4·25-s − 26-s + 4·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.34·5-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1.54·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.436·21-s + 1.06·22-s − 1.87·23-s − 1.22·24-s + 4/5·25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 61 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−14.73281781004713716800542459589, −13.13927303138002944840689395149, −11.99765290224431389063190901976, −11.01835010861511067157429624329, −10.14754439690118852610085973818, −8.287570693733701613432166498970, −7.66039763333771123227596439320, −5.60157612088078313975836309422, −4.25666915587159511841100490901, 0,
4.25666915587159511841100490901, 5.60157612088078313975836309422, 7.66039763333771123227596439320, 8.287570693733701613432166498970, 10.14754439690118852610085973818, 11.01835010861511067157429624329, 11.99765290224431389063190901976, 13.13927303138002944840689395149, 14.73281781004713716800542459589