| L(s) = 1 | − 2·11-s − 16-s − 2·17-s − 2·23-s − 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·49-s − 2·71-s + 2·73-s − 2·79-s − 81-s + 2·89-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·11-s − 16-s − 2·17-s − 2·23-s − 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·49-s − 2·71-s + 2·73-s − 2·79-s − 81-s + 2·89-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5020002568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5020002568\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 233 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−9.024199886102955549724432401123, −8.966308471757042360398399669638, −8.531493529962721411274021462105, −8.198120055933097180651501151626, −7.56992681696179587969796202837, −7.39169254687569231610171592698, −7.24586468610773280239023326767, −6.50892096593132392100438424575, −6.03133522837937432936503459265, −5.97810237262983055390819926905, −5.28800620688109138976054973666, −5.02538466275586459080282852221, −4.47807619637250736327595134535, −4.02889128610351794862898201543, −3.91096781329385457098304243135, −2.88022517795644054179448284056, −2.64741712167278805447573544247, −2.01167708205642339916872271798, −1.94669311806842822348200736691, −0.42656929302875800099008565060,
0.42656929302875800099008565060, 1.94669311806842822348200736691, 2.01167708205642339916872271798, 2.64741712167278805447573544247, 2.88022517795644054179448284056, 3.91096781329385457098304243135, 4.02889128610351794862898201543, 4.47807619637250736327595134535, 5.02538466275586459080282852221, 5.28800620688109138976054973666, 5.97810237262983055390819926905, 6.03133522837937432936503459265, 6.50892096593132392100438424575, 7.24586468610773280239023326767, 7.39169254687569231610171592698, 7.56992681696179587969796202837, 8.198120055933097180651501151626, 8.531493529962721411274021462105, 8.966308471757042360398399669638, 9.024199886102955549724432401123
