L(s) = 1 | − 0.849·2-s − 3-s − 1.27·4-s − 2.48·5-s + 0.849·6-s + 1.35·7-s + 2.78·8-s + 9-s + 2.11·10-s − 11-s + 1.27·12-s − 1.14·14-s + 2.48·15-s + 0.194·16-s − 2.42·17-s − 0.849·18-s − 0.806·19-s + 3.17·20-s − 1.35·21-s + 0.849·22-s − 2.92·23-s − 2.78·24-s + 1.17·25-s − 27-s − 1.72·28-s + 8.28·29-s − 2.11·30-s + ⋯ |
L(s) = 1 | − 0.600·2-s − 0.577·3-s − 0.639·4-s − 1.11·5-s + 0.346·6-s + 0.510·7-s + 0.984·8-s + 0.333·9-s + 0.667·10-s − 0.301·11-s + 0.369·12-s − 0.306·14-s + 0.641·15-s + 0.0485·16-s − 0.587·17-s − 0.200·18-s − 0.185·19-s + 0.710·20-s − 0.294·21-s + 0.181·22-s − 0.609·23-s − 0.568·24-s + 0.235·25-s − 0.192·27-s − 0.326·28-s + 1.53·29-s − 0.385·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.849T + 2T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 + 0.806T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 5.14T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 - 7.73T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 - 3.70T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 1.84T + 89T^{2} \) |
| 97 | \( 1 + 0.284T + 97T^{2} \) |
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
−7.996653011366227426258811703534, −7.22466718878483064931880175595, −6.55447280661159950872633549984, −5.46681875932353935734201082148, −4.76327952570232624441621744776, −4.25603747428403513371741761515, −3.47980130970615359816006590004, −2.11406696296252633151124575644, −0.924512800842892023118110063063, 0,
0.924512800842892023118110063063, 2.11406696296252633151124575644, 3.47980130970615359816006590004, 4.25603747428403513371741761515, 4.76327952570232624441621744776, 5.46681875932353935734201082148, 6.55447280661159950872633549984, 7.22466718878483064931880175595, 7.996653011366227426258811703534