| L(s) = 1 | − 2·5-s − 2·9-s − 8·13-s − 6·17-s − 7·25-s − 4·29-s + 20·37-s + 20·41-s + 4·45-s − 5·49-s − 8·53-s − 26·61-s + 16·65-s + 18·73-s − 5·81-s + 12·85-s + 24·89-s − 16·97-s − 20·101-s − 20·113-s + 16·117-s + 3·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s − 2.21·13-s − 1.45·17-s − 7/5·25-s − 0.742·29-s + 3.28·37-s + 3.12·41-s + 0.596·45-s − 5/7·49-s − 1.09·53-s − 3.32·61-s + 1.98·65-s + 2.10·73-s − 5/9·81-s + 1.30·85-s + 2.54·89-s − 1.62·97-s − 1.99·101-s − 1.88·113-s + 1.47·117-s + 3/11·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.68233725674896618441562917388, −9.569234090496287600010900905430, −9.553328544165326093753901438762, −9.084561409834522310688102275122, −7.982276472778928545712318850832, −7.62429229152375892340957919668, −7.61880161741027781839162381092, −6.44323514092267885981017350010, −6.06650473039374929409046448898, −5.18067831877099909482237924161, −4.37905347680940090915441446169, −4.15537276955970084853193543262, −2.82511261070703297705797963037, −2.31923248203641562194456416521, 0,
2.31923248203641562194456416521, 2.82511261070703297705797963037, 4.15537276955970084853193543262, 4.37905347680940090915441446169, 5.18067831877099909482237924161, 6.06650473039374929409046448898, 6.44323514092267885981017350010, 7.61880161741027781839162381092, 7.62429229152375892340957919668, 7.982276472778928545712318850832, 9.084561409834522310688102275122, 9.553328544165326093753901438762, 9.569234090496287600010900905430, 10.68233725674896618441562917388
