| L(s) = 1 | − 4·7-s + 9-s − 4·17-s − 12·23-s + 25-s + 4·41-s − 4·47-s + 2·49-s − 4·63-s − 8·71-s − 12·73-s − 8·79-s + 81-s − 12·89-s − 12·97-s + 4·103-s + 12·113-s + 16·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1/3·9-s − 0.970·17-s − 2.50·23-s + 1/5·25-s + 0.624·41-s − 0.583·47-s + 2/7·49-s − 0.503·63-s − 0.949·71-s − 1.40·73-s − 0.900·79-s + 1/9·81-s − 1.27·89-s − 1.21·97-s + 0.394·103-s + 1.12·113-s + 1.46·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−9.789558936963360670762164057052, −9.432331843962428674696661297451, −8.651227420043199698826817811666, −8.349861868559778434364327135898, −7.58552861389566241021822241490, −7.08242127393491403183760755045, −6.54113518303047890819319823827, −6.03142414243910032413180227612, −5.69872404151448590531497577728, −4.58279675140756191403136982971, −4.17610479998371727321127812323, −3.45201131090447634672216722360, −2.73019489142800131650849040974, −1.82414167365812974415618269325, 0,
1.82414167365812974415618269325, 2.73019489142800131650849040974, 3.45201131090447634672216722360, 4.17610479998371727321127812323, 4.58279675140756191403136982971, 5.69872404151448590531497577728, 6.03142414243910032413180227612, 6.54113518303047890819319823827, 7.08242127393491403183760755045, 7.58552861389566241021822241490, 8.349861868559778434364327135898, 8.651227420043199698826817811666, 9.432331843962428674696661297451, 9.789558936963360670762164057052
