| L(s) = 1 | + 3-s + 2·4-s + 5-s + 11-s + 2·12-s − 13-s + 15-s + 3·16-s − 17-s + 2·20-s − 27-s − 2·29-s + 33-s − 39-s + 2·44-s + 2·47-s + 3·48-s − 51-s − 2·52-s + 55-s + 2·60-s + 4·64-s − 65-s − 2·68-s − 2·71-s − 73-s − 2·79-s + ⋯ |
| L(s) = 1 | + 3-s + 2·4-s + 5-s + 11-s + 2·12-s − 13-s + 15-s + 3·16-s − 17-s + 2·20-s − 27-s − 2·29-s + 33-s − 39-s + 2·44-s + 2·47-s + 3·48-s − 51-s − 2·52-s + 55-s + 2·60-s + 4·64-s − 65-s − 2·68-s − 2·71-s − 73-s − 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.457191641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457191641\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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
−9.461649075383412604685670395989, −8.933404997016142708570739941536, −8.830634426764261253818250608788, −8.291246865939892314338234982898, −7.70542270478423696039592675332, −7.40442447092486514667129213305, −7.13084045293438558390788670561, −6.93219664597148579633567750759, −6.21716041625704478195306475335, −6.00636610709355256618885161749, −5.64593344389995164486883232500, −5.37423098090952954652379940149, −4.41722737769005268360835949646, −4.11143142151056214534921320114, −3.48998316761678450915348268612, −2.97826453295927527026953194208, −2.66113838759954693414542334677, −2.00982462259905650845547969586, −1.98867027134210920072959217847, −1.36147802593647645543938663821,
1.36147802593647645543938663821, 1.98867027134210920072959217847, 2.00982462259905650845547969586, 2.66113838759954693414542334677, 2.97826453295927527026953194208, 3.48998316761678450915348268612, 4.11143142151056214534921320114, 4.41722737769005268360835949646, 5.37423098090952954652379940149, 5.64593344389995164486883232500, 6.00636610709355256618885161749, 6.21716041625704478195306475335, 6.93219664597148579633567750759, 7.13084045293438558390788670561, 7.40442447092486514667129213305, 7.70542270478423696039592675332, 8.291246865939892314338234982898, 8.830634426764261253818250608788, 8.933404997016142708570739941536, 9.461649075383412604685670395989
