| L(s) = 1 | + 3·4-s + 4·5-s + 11-s + 5·16-s + 12·20-s + 11·25-s − 8·31-s + 3·44-s + 10·49-s + 4·55-s + 8·59-s + 3·64-s + 20·80-s + 33·100-s + 121-s − 24·124-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.78·5-s + 0.301·11-s + 5/4·16-s + 2.68·20-s + 11/5·25-s − 1.43·31-s + 0.452·44-s + 10/7·49-s + 0.539·55-s + 1.04·59-s + 3/8·64-s + 2.23·80-s + 3.29·100-s + 1/11·121-s − 2.15·124-s + 2.14·125-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 − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.733368776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.733368776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−7.44071012856179454174115168498, −6.98461506866828385433924515992, −6.84884806265829454446157697534, −6.30325696612682555597887492081, −6.02499817336897508665224642231, −5.48878341540355437707060982436, −5.42457431972828552322534326719, −4.72801673816053907358368854207, −4.05560224434393851290900259231, −3.51309106430050388189908891790, −2.87537560535593228966647235605, −2.51187495471372643516646191681, −1.93873899637775283933678594487, −1.69569533896967354908280090098, −0.912271776203355030874824007294,
0.912271776203355030874824007294, 1.69569533896967354908280090098, 1.93873899637775283933678594487, 2.51187495471372643516646191681, 2.87537560535593228966647235605, 3.51309106430050388189908891790, 4.05560224434393851290900259231, 4.72801673816053907358368854207, 5.42457431972828552322534326719, 5.48878341540355437707060982436, 6.02499817336897508665224642231, 6.30325696612682555597887492081, 6.84884806265829454446157697534, 6.98461506866828385433924515992, 7.44071012856179454174115168498
