| L(s) = 1 | + 2·9-s − 25-s + 4·41-s − 2·49-s + 3·81-s − 4·89-s − 2·121-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 + 211-s + 223-s − 2·225-s + ⋯ |
| L(s) = 1 | + 2·9-s − 25-s + 4·41-s − 2·49-s + 3·81-s − 4·89-s − 2·121-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 + 211-s + 223-s − 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.280823156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280823156\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 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
−9.952380009061971476016893250330, −9.667459165055266968595816369154, −9.362415640710904540750174732075, −9.081758883693930174934303337483, −8.246088991136250399594171153945, −8.064006775761831723103436634520, −7.51041393700793237534338313363, −7.33608252330103242552435231391, −6.83982491707451663541364549824, −6.39194979146782392669591705024, −5.93330150882845923892834145932, −5.54359395567518135212106836468, −4.82352186882573125275769387637, −4.50307621547822240437210255771, −3.97423021975834343415153124354, −3.82502464749591129704786151542, −2.89733515607916253952647196850, −2.38980190813354691460753359718, −1.64330031213782616999717196204, −1.12985572546244192274549107844,
1.12985572546244192274549107844, 1.64330031213782616999717196204, 2.38980190813354691460753359718, 2.89733515607916253952647196850, 3.82502464749591129704786151542, 3.97423021975834343415153124354, 4.50307621547822240437210255771, 4.82352186882573125275769387637, 5.54359395567518135212106836468, 5.93330150882845923892834145932, 6.39194979146782392669591705024, 6.83982491707451663541364549824, 7.33608252330103242552435231391, 7.51041393700793237534338313363, 8.064006775761831723103436634520, 8.246088991136250399594171153945, 9.081758883693930174934303337483, 9.362415640710904540750174732075, 9.667459165055266968595816369154, 9.952380009061971476016893250330
