| L(s) = 1 | + 3-s − 4·4-s + 4·7-s + 9-s − 4·12-s − 8·13-s + 12·16-s + 4·21-s − 5·25-s + 27-s − 16·28-s + 10·31-s − 4·36-s − 8·39-s + 12·48-s − 2·49-s + 32·52-s + 10·61-s + 4·63-s − 32·64-s + 4·67-s − 5·75-s + 81-s − 16·84-s − 32·91-s + 10·93-s + 16·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s + 1.51·7-s + 1/3·9-s − 1.15·12-s − 2.21·13-s + 3·16-s + 0.872·21-s − 25-s + 0.192·27-s − 3.02·28-s + 1.79·31-s − 2/3·36-s − 1.28·39-s + 1.73·48-s − 2/7·49-s + 4.43·52-s + 1.28·61-s + 0.503·63-s − 4·64-s + 0.488·67-s − 0.577·75-s + 1/9·81-s − 1.74·84-s − 3.35·91-s + 1.03·93-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319117816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319117816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 41 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−8.196345431397680357302796482250, −7.79233496370511093882409008363, −7.49403265627965125182213923160, −6.91578123961861233078487193839, −6.20673562115283206405918215507, −5.58589920207151454855958169522, −5.08454827023519235113594175171, −4.83383633012443014176031867945, −4.56873148096175944273786655950, −4.07775735531544790586167122014, −3.53596063438026803301590958362, −2.80227898078716634470716031538, −2.20402583524777241101161396968, −1.45483538337878128634228710026, −0.53956358269126821379536745804,
0.53956358269126821379536745804, 1.45483538337878128634228710026, 2.20402583524777241101161396968, 2.80227898078716634470716031538, 3.53596063438026803301590958362, 4.07775735531544790586167122014, 4.56873148096175944273786655950, 4.83383633012443014176031867945, 5.08454827023519235113594175171, 5.58589920207151454855958169522, 6.20673562115283206405918215507, 6.91578123961861233078487193839, 7.49403265627965125182213923160, 7.79233496370511093882409008363, 8.196345431397680357302796482250
