| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 9-s − 3·12-s + 5·16-s + 2·18-s − 19-s − 4·24-s + 3·25-s − 27-s + 5·29-s + 6·32-s + 3·36-s − 2·38-s − 7·41-s − 43-s − 5·48-s − 5·49-s + 6·50-s + 25·53-s − 2·54-s + 57-s + 10·58-s − 3·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.41·8-s + 1/3·9-s − 0.866·12-s + 5/4·16-s + 0.471·18-s − 0.229·19-s − 0.816·24-s + 3/5·25-s − 0.192·27-s + 0.928·29-s + 1.06·32-s + 1/2·36-s − 0.324·38-s − 1.09·41-s − 0.152·43-s − 0.721·48-s − 5/7·49-s + 0.848·50-s + 3.43·53-s − 0.272·54-s + 0.132·57-s + 1.31·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.299880458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.299880458\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 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
−8.361265672128935441811174080999, −7.48859481061686387107035219065, −7.30587353278051202258048014272, −6.70767582860030222237756821121, −6.48686802846743604621087881709, −5.92373873817053490306710474889, −5.45957676511972144815075201256, −5.11024117639995378664900312487, −4.58928450255563018818355364692, −4.18068823964036723605526602796, −3.62024639011733655369904103227, −3.06659408609913902153102982556, −2.44956854690341153540879083012, −1.79338212263893682557521333205, −0.855990091492211522246527594829,
0.855990091492211522246527594829, 1.79338212263893682557521333205, 2.44956854690341153540879083012, 3.06659408609913902153102982556, 3.62024639011733655369904103227, 4.18068823964036723605526602796, 4.58928450255563018818355364692, 5.11024117639995378664900312487, 5.45957676511972144815075201256, 5.92373873817053490306710474889, 6.48686802846743604621087881709, 6.70767582860030222237756821121, 7.30587353278051202258048014272, 7.48859481061686387107035219065, 8.361265672128935441811174080999
