| L(s) = 1 | + 3·3-s + 5·7-s + 6·9-s + 15·19-s + 15·21-s + 9·27-s − 18·31-s + 37-s + 16·43-s + 18·49-s + 45·57-s − 27·61-s + 30·63-s + 5·67-s + 27·73-s + 13·79-s + 9·81-s − 54·93-s − 33·103-s − 19·109-s + 3·111-s − 11·121-s + 127-s + 48·129-s + 131-s + 75·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.88·7-s + 2·9-s + 3.44·19-s + 3.27·21-s + 1.73·27-s − 3.23·31-s + 0.164·37-s + 2.43·43-s + 18/7·49-s + 5.96·57-s − 3.45·61-s + 3.77·63-s + 0.610·67-s + 3.16·73-s + 1.46·79-s + 81-s − 5.59·93-s − 3.25·103-s − 1.81·109-s + 0.284·111-s − 121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 6.50·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.083715581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.083715581\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.290814512440635611877066157263, −9.215293485749154785716883046137, −8.269162957708945794247702833254, −8.197559621307013370347202214462, −7.69745725982321426073260772789, −7.60193180788418036503028117559, −7.14390614218787308485474263900, −7.03763539329666039795409260529, −6.00551208795012122467782802700, −5.58475424862193069708378679908, −5.10429327590881580721646497541, −5.08126954872234791849929113241, −4.12165593101399199337932657434, −4.08591098968372334681320559963, −3.24766862070557799029412032541, −3.23656042104179784030453871975, −2.37247346927163917509529778181, −2.02668303173117687912630079575, −1.37978528764240908399716322552, −1.04006884104734251891020201187,
1.04006884104734251891020201187, 1.37978528764240908399716322552, 2.02668303173117687912630079575, 2.37247346927163917509529778181, 3.23656042104179784030453871975, 3.24766862070557799029412032541, 4.08591098968372334681320559963, 4.12165593101399199337932657434, 5.08126954872234791849929113241, 5.10429327590881580721646497541, 5.58475424862193069708378679908, 6.00551208795012122467782802700, 7.03763539329666039795409260529, 7.14390614218787308485474263900, 7.60193180788418036503028117559, 7.69745725982321426073260772789, 8.197559621307013370347202214462, 8.269162957708945794247702833254, 9.215293485749154785716883046137, 9.290814512440635611877066157263
