| L(s) = 1 | + 9-s + 2·11-s − 2·19-s − 2·41-s − 2·49-s + 4·59-s + 2·89-s + 2·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 9-s + 2·11-s − 2·19-s − 2·41-s − 2·49-s + 4·59-s + 2·89-s + 2·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.048754437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048754437\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$ | \( ( 1 - T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.48192462057179431669655267948, −10.26471179692433523487791927714, −9.888970010120026042719453466201, −9.433726876346548985088267965846, −8.974148495807358145771356534967, −8.524658810398930347786710734400, −8.360859442106513473343411294213, −7.66976433374811804530926445176, −7.05938144605083630950688159385, −6.76979535282880298690896417891, −6.30808991530566763121933921838, −6.24893824838174372412763924405, −5.13329225274981955757854630763, −4.96042355551600179416120835091, −4.14552487292280802101351485904, −3.89058780516858376452091421139, −3.54888068522024869069231136553, −2.49032747626348307530879777677, −1.84068117969644956727528700491, −1.27683073765345448383954374058,
1.27683073765345448383954374058, 1.84068117969644956727528700491, 2.49032747626348307530879777677, 3.54888068522024869069231136553, 3.89058780516858376452091421139, 4.14552487292280802101351485904, 4.96042355551600179416120835091, 5.13329225274981955757854630763, 6.24893824838174372412763924405, 6.30808991530566763121933921838, 6.76979535282880298690896417891, 7.05938144605083630950688159385, 7.66976433374811804530926445176, 8.360859442106513473343411294213, 8.524658810398930347786710734400, 8.974148495807358145771356534967, 9.433726876346548985088267965846, 9.888970010120026042719453466201, 10.26471179692433523487791927714, 10.48192462057179431669655267948
