| L(s) = 1 | − 2·7-s + 9-s − 2·17-s + 25-s − 4·31-s + 2·47-s + 49-s − 2·63-s − 2·71-s + 4·113-s + 4·119-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·7-s + 9-s − 2·17-s + 25-s − 4·31-s + 2·47-s + 49-s − 2·63-s − 2·71-s + 4·113-s + 4·119-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6250707074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6250707074\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.163644565725762270798183418088, −8.857798240330498516345391000126, −8.456659332208440573307647116278, −7.55995553856146921579089603365, −7.44208257964145552273456310161, −7.06980072597460432043684858967, −6.87962826967830132674995307973, −6.34227980712601482043790375220, −6.19298973486901116145811799948, −5.57766944819752483179688164989, −5.34318332488671053338281874127, −4.61457999541250677639629245388, −4.37153898682535228787541908803, −3.73291840204906452544089757887, −3.69267487646518361871695322462, −3.01423974435802434818444476930, −2.63102501446484539378531365409, −1.96319392363277027465208035418, −1.60575815015754326078566069670, −0.45468294299043555927308031050,
0.45468294299043555927308031050, 1.60575815015754326078566069670, 1.96319392363277027465208035418, 2.63102501446484539378531365409, 3.01423974435802434818444476930, 3.69267487646518361871695322462, 3.73291840204906452544089757887, 4.37153898682535228787541908803, 4.61457999541250677639629245388, 5.34318332488671053338281874127, 5.57766944819752483179688164989, 6.19298973486901116145811799948, 6.34227980712601482043790375220, 6.87962826967830132674995307973, 7.06980072597460432043684858967, 7.44208257964145552273456310161, 7.55995553856146921579089603365, 8.456659332208440573307647116278, 8.857798240330498516345391000126, 9.163644565725762270798183418088
