| L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·13-s + 19-s + 21-s + 25-s + 5·27-s + 4·31-s − 9·37-s − 3·39-s − 2·43-s − 11·49-s − 57-s + 9·61-s + 2·63-s + 6·67-s − 11·73-s − 75-s − 79-s + 81-s − 3·91-s − 4·93-s − 6·97-s − 21·103-s + 7·109-s + 9·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.832·13-s + 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.962·27-s + 0.718·31-s − 1.47·37-s − 0.480·39-s − 0.304·43-s − 1.57·49-s − 0.132·57-s + 1.15·61-s + 0.251·63-s + 0.733·67-s − 1.28·73-s − 0.115·75-s − 0.112·79-s + 1/9·81-s − 0.314·91-s − 0.414·93-s − 0.609·97-s − 2.06·103-s + 0.670·109-s + 0.854·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.056123480365903629043575681891, −7.21362966648318190392259676276, −6.83047795373991082531197117061, −6.59380768935374630359400587145, −6.00160142021858507016706586024, −5.66297641672445603367551367483, −5.25410869956917285448236746985, −4.72436668358284651272578043350, −4.23241424827694953808817331754, −3.48007479496248255497641137232, −3.23056133880132485069937199939, −2.58569471700980132070684929029, −1.78128567945817102845343504613, −1.00250451570177790120488055064, 0,
1.00250451570177790120488055064, 1.78128567945817102845343504613, 2.58569471700980132070684929029, 3.23056133880132485069937199939, 3.48007479496248255497641137232, 4.23241424827694953808817331754, 4.72436668358284651272578043350, 5.25410869956917285448236746985, 5.66297641672445603367551367483, 6.00160142021858507016706586024, 6.59380768935374630359400587145, 6.83047795373991082531197117061, 7.21362966648318190392259676276, 8.056123480365903629043575681891
