| L(s) = 1 | − 2·5-s − 6·9-s − 25-s − 20·29-s + 20·41-s + 12·45-s − 14·49-s − 20·61-s + 27·81-s + 20·89-s − 4·101-s + 12·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s − 1/5·25-s − 3.71·29-s + 3.12·41-s + 1.78·45-s − 2·49-s − 2.56·61-s + 3·81-s + 2.11·89-s − 0.398·101-s + 1.14·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.84453522425202397161354104633, −9.617316516935557446365175892726, −9.206579341791145473157585005601, −8.955386231165229198073332132052, −8.025782606763976976877695514876, −7.77199473906097062385997282225, −7.38383189330491254934984805094, −6.27997548582365804353754428543, −5.87146418848833687506982135026, −5.37327377173791634385808352079, −4.46989807490289628768060263585, −3.67478222653086463350186782835, −3.13241440937412885242101141742, −2.12458086750050120053975349819, 0,
2.12458086750050120053975349819, 3.13241440937412885242101141742, 3.67478222653086463350186782835, 4.46989807490289628768060263585, 5.37327377173791634385808352079, 5.87146418848833687506982135026, 6.27997548582365804353754428543, 7.38383189330491254934984805094, 7.77199473906097062385997282225, 8.025782606763976976877695514876, 8.955386231165229198073332132052, 9.206579341791145473157585005601, 9.617316516935557446365175892726, 10.84453522425202397161354104633
