| L(s) = 1 | + 4·5-s − 4·13-s − 8·17-s + 2·25-s + 12·29-s + 4·37-s − 24·41-s − 10·49-s + 12·53-s + 20·61-s − 16·65-s + 4·73-s − 32·85-s + 16·89-s − 4·97-s + 12·101-s − 36·109-s − 16·113-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.10·13-s − 1.94·17-s + 2/5·25-s + 2.22·29-s + 0.657·37-s − 3.74·41-s − 1.42·49-s + 1.64·53-s + 2.56·61-s − 1.98·65-s + 0.468·73-s − 3.47·85-s + 1.69·89-s − 0.406·97-s + 1.19·101-s − 3.44·109-s − 1.50·113-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−7.84652290203924509416522386179, −7.07290137067261705026994657338, −6.62751242713174496166735466724, −6.62630504499756739471306360707, −6.15420646659294641678088585030, −5.39382132947189393802299472560, −5.14932968174065587920886153262, −4.82152978324827041241210558809, −4.19902739124228802218338857400, −3.59372732811385513436205218547, −2.78285477570770474698242693497, −2.28896400528402159654461092906, −2.07793680595226141750788729163, −1.27289759970492945430986166195, 0,
1.27289759970492945430986166195, 2.07793680595226141750788729163, 2.28896400528402159654461092906, 2.78285477570770474698242693497, 3.59372732811385513436205218547, 4.19902739124228802218338857400, 4.82152978324827041241210558809, 5.14932968174065587920886153262, 5.39382132947189393802299472560, 6.15420646659294641678088585030, 6.62630504499756739471306360707, 6.62751242713174496166735466724, 7.07290137067261705026994657338, 7.84652290203924509416522386179
