L(s) = 1 | − 9-s − 8·11-s − 8·19-s − 12·29-s − 16·31-s − 12·41-s + 14·49-s + 8·59-s − 4·61-s − 16·71-s − 16·79-s + 81-s + 12·89-s + 8·99-s − 36·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 2.41·11-s − 1.83·19-s − 2.22·29-s − 2.87·31-s − 1.87·41-s + 2·49-s + 1.04·59-s − 0.512·61-s − 1.89·71-s − 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 3.58·101-s + 0.383·109-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.442517922138744862066974570191, −9.095569430131749971545586709374, −8.681755263756113727855447818436, −8.421824338907487617689049402103, −7.73362334914342770735122976781, −7.65405360595228567799146346241, −7.10487299799159393557263403488, −6.80311028026961086387750024953, −6.00872006837976967030667424282, −5.58338788664374810301843404421, −5.39975766186818179612635171872, −5.05554635793203841842230107205, −4.09791195080334812116643820526, −4.03478677402135871280048824257, −3.21109854637721406124469186890, −2.73704710677610881646288420532, −2.04555220160031712880904490069, −1.80509395194441917922238338027, 0, 0,
1.80509395194441917922238338027, 2.04555220160031712880904490069, 2.73704710677610881646288420532, 3.21109854637721406124469186890, 4.03478677402135871280048824257, 4.09791195080334812116643820526, 5.05554635793203841842230107205, 5.39975766186818179612635171872, 5.58338788664374810301843404421, 6.00872006837976967030667424282, 6.80311028026961086387750024953, 7.10487299799159393557263403488, 7.65405360595228567799146346241, 7.73362334914342770735122976781, 8.421824338907487617689049402103, 8.681755263756113727855447818436, 9.095569430131749971545586709374, 9.442517922138744862066974570191