| L(s) = 1 | + 9-s + 2·23-s + 25-s + 4·71-s − 2·79-s − 4·113-s − 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 + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 2·207-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 9-s + 2·23-s + 25-s + 4·71-s − 2·79-s − 4·113-s − 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 + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 2·207-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.350758579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350758579\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 - T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.694381725792712600167187978976, −9.464044145942750369160035060901, −8.923318284569698826753774160929, −8.883228749720093051272450253407, −8.027582556049617378691482669489, −7.996165296196794000188759385747, −7.36075359358175247559203255674, −6.94877367630309220069908736610, −6.56094718757835476748542893622, −6.54268735658067841290736827570, −5.45028950423301394584583713214, −5.42567690270857025392685933902, −4.78824312014541780773161634249, −4.52754966427191602893271009139, −3.75521902708915628478982505982, −3.60515802224397829403611710894, −2.64327215633069232308627164410, −2.57669663542371107623774449255, −1.45717556660701804980797676173, −1.09811358507326840712078854483,
1.09811358507326840712078854483, 1.45717556660701804980797676173, 2.57669663542371107623774449255, 2.64327215633069232308627164410, 3.60515802224397829403611710894, 3.75521902708915628478982505982, 4.52754966427191602893271009139, 4.78824312014541780773161634249, 5.42567690270857025392685933902, 5.45028950423301394584583713214, 6.54268735658067841290736827570, 6.56094718757835476748542893622, 6.94877367630309220069908736610, 7.36075359358175247559203255674, 7.996165296196794000188759385747, 8.027582556049617378691482669489, 8.883228749720093051272450253407, 8.923318284569698826753774160929, 9.464044145942750369160035060901, 9.694381725792712600167187978976
