| L(s) = 1 | + 2.13e3·5-s + 1.68e4·7-s + 7.04e5·11-s + 9.52e5·13-s − 5.10e6·17-s − 1.39e7·19-s + 1.89e7·23-s − 4.42e7·25-s − 1.48e8·29-s − 1.59e8·31-s + 3.57e7·35-s − 8.25e7·37-s + 7.29e8·41-s + 1.18e9·43-s − 2.86e8·47-s + 2.82e8·49-s − 3.85e9·53-s + 1.49e9·55-s − 5.28e9·59-s − 8.15e9·61-s + 2.02e9·65-s + 9.25e9·67-s − 2.00e10·71-s + 2.38e10·73-s + 1.18e10·77-s + 3.51e9·79-s + 2.14e10·83-s + ⋯ |
| L(s) = 1 | + 0.304·5-s + 0.377·7-s + 1.31·11-s + 0.711·13-s − 0.872·17-s − 1.28·19-s + 0.613·23-s − 0.907·25-s − 1.34·29-s − 0.998·31-s + 0.115·35-s − 0.195·37-s + 0.983·41-s + 1.22·43-s − 0.181·47-s + 1/7·49-s − 1.26·53-s + 0.401·55-s − 0.963·59-s − 1.23·61-s + 0.216·65-s + 0.837·67-s − 1.31·71-s + 1.34·73-s + 0.498·77-s + 0.128·79-s + 0.599·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 426 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 704196 T + p^{11} T^{2} \) |
| 13 | \( 1 - 952286 T + p^{11} T^{2} \) |
| 17 | \( 1 + 5105514 T + p^{11} T^{2} \) |
| 19 | \( 1 + 13905148 T + p^{11} T^{2} \) |
| 23 | \( 1 - 18945408 T + p^{11} T^{2} \) |
| 29 | \( 1 + 148937094 T + p^{11} T^{2} \) |
| 31 | \( 1 + 159226144 T + p^{11} T^{2} \) |
| 37 | \( 1 + 82548178 T + p^{11} T^{2} \) |
| 41 | \( 1 - 729417150 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1185139028 T + p^{11} T^{2} \) |
| 47 | \( 1 + 286058928 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3853540014 T + p^{11} T^{2} \) |
| 59 | \( 1 + 5288267196 T + p^{11} T^{2} \) |
| 61 | \( 1 + 8156327602 T + p^{11} T^{2} \) |
| 67 | \( 1 - 9250048316 T + p^{11} T^{2} \) |
| 71 | \( 1 + 20051655792 T + p^{11} T^{2} \) |
| 73 | \( 1 - 23853193802 T + p^{11} T^{2} \) |
| 79 | \( 1 - 3513675584 T + p^{11} T^{2} \) |
| 83 | \( 1 - 21497352012 T + p^{11} T^{2} \) |
| 89 | \( 1 + 66839945634 T + p^{11} T^{2} \) |
| 97 | \( 1 - 146492724002 T + p^{11} T^{2} \) |
| 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
−9.352246845069877287640552881366, −8.910672125671937576845578933371, −7.69504301847692952963252882426, −6.56234780018984140500060394916, −5.83206671855053102816213312379, −4.45168562517235053739785194550, −3.67568081391644812208881531697, −2.14347869546164577084595817277, −1.36119925682415161896920069780, 0,
1.36119925682415161896920069780, 2.14347869546164577084595817277, 3.67568081391644812208881531697, 4.45168562517235053739785194550, 5.83206671855053102816213312379, 6.56234780018984140500060394916, 7.69504301847692952963252882426, 8.910672125671937576845578933371, 9.352246845069877287640552881366
