| L(s) = 1 | − 2·3-s + 9-s − 4·13-s + 12·23-s + 4·27-s − 4·37-s + 8·39-s − 12·47-s − 10·49-s − 24·59-s + 4·61-s − 24·69-s + 24·71-s − 4·73-s − 11·81-s + 12·83-s − 4·97-s − 12·107-s + 4·109-s + 8·111-s − 4·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 24·141-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.10·13-s + 2.50·23-s + 0.769·27-s − 0.657·37-s + 1.28·39-s − 1.75·47-s − 1.42·49-s − 3.12·59-s + 0.512·61-s − 2.88·69-s + 2.84·71-s − 0.468·73-s − 1.22·81-s + 1.31·83-s − 0.406·97-s − 1.16·107-s + 0.383·109-s + 0.759·111-s − 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.02·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.407676785947288187996164912905, −8.008294857926527437201173134092, −7.47121424818625422471458115292, −6.96167091691043492420755836355, −6.43307814142801241380910059979, −6.41842389830764202453884824049, −5.30602566274475700528497955431, −5.24854435453179347418331807133, −4.82118726362395324834121081499, −4.28530365432405873873397666950, −3.16807967427295861033734664069, −3.07875775618643284846845606551, −2.01836233216080453835298575952, −1.10363850080882238424508077779, 0,
1.10363850080882238424508077779, 2.01836233216080453835298575952, 3.07875775618643284846845606551, 3.16807967427295861033734664069, 4.28530365432405873873397666950, 4.82118726362395324834121081499, 5.24854435453179347418331807133, 5.30602566274475700528497955431, 6.41842389830764202453884824049, 6.43307814142801241380910059979, 6.96167091691043492420755836355, 7.47121424818625422471458115292, 8.008294857926527437201173134092, 8.407676785947288187996164912905
