| L(s) = 1 | − 2·4-s − 6·5-s + 5·9-s + 4·16-s + 12·20-s + 17·25-s − 10·36-s − 30·45-s − 14·49-s − 8·64-s − 24·80-s + 16·81-s − 34·100-s − 11·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 2.68·5-s + 5/3·9-s + 16-s + 2.68·20-s + 17/5·25-s − 5/3·36-s − 4.47·45-s − 2·49-s − 64-s − 2.68·80-s + 16/9·81-s − 3.39·100-s − 121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 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
−7.980431429152566769157900669073, −7.78200973201510429323459130616, −7.37245526981052883338498731623, −6.93923637491770959275076371341, −6.51153196902546005149258982097, −5.74927243615175067187714306744, −4.96947418530356284782006198809, −4.69316882482002854566469909847, −4.23736927114502957398282007479, −3.92970967637566964233616740538, −3.51121938611793196415265051944, −3.01352411256357864770761039200, −1.74114849414903707672454341596, −0.870792558864617165022266413742, 0,
0.870792558864617165022266413742, 1.74114849414903707672454341596, 3.01352411256357864770761039200, 3.51121938611793196415265051944, 3.92970967637566964233616740538, 4.23736927114502957398282007479, 4.69316882482002854566469909847, 4.96947418530356284782006198809, 5.74927243615175067187714306744, 6.51153196902546005149258982097, 6.93923637491770959275076371341, 7.37245526981052883338498731623, 7.78200973201510429323459130616, 7.980431429152566769157900669073
