| L(s) = 1 | − 2·3-s − 4·4-s + 2·7-s + 9-s + 8·12-s + 2·13-s + 12·16-s − 14·19-s − 4·21-s − 25-s + 4·27-s − 8·28-s + 10·31-s − 4·36-s + 4·37-s − 4·39-s − 2·43-s − 24·48-s + 3·49-s − 8·52-s + 28·57-s − 20·61-s + 2·63-s − 32·64-s + 28·67-s + 22·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s + 0.755·7-s + 1/3·9-s + 2.30·12-s + 0.554·13-s + 3·16-s − 3.21·19-s − 0.872·21-s − 1/5·25-s + 0.769·27-s − 1.51·28-s + 1.79·31-s − 2/3·36-s + 0.657·37-s − 0.640·39-s − 0.304·43-s − 3.46·48-s + 3/7·49-s − 1.10·52-s + 3.70·57-s − 2.56·61-s + 0.251·63-s − 4·64-s + 3.42·67-s + 2.57·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.542604600630706787025977559837, −8.965474064429713311069674592028, −8.524079546582096676065943493512, −8.023034081311283559976527256533, −8.011832969610721686895020035599, −6.49992505927329333857395184343, −6.48432181419363077598714474936, −5.79128008848021850482104511614, −4.95454046107923865998450494116, −4.94336211753658685120988545492, −4.04609643315305459582542236808, −3.97318584788109067523740792933, −2.53593630462827092595495233291, −1.17094069297371334537206633547, 0,
1.17094069297371334537206633547, 2.53593630462827092595495233291, 3.97318584788109067523740792933, 4.04609643315305459582542236808, 4.94336211753658685120988545492, 4.95454046107923865998450494116, 5.79128008848021850482104511614, 6.48432181419363077598714474936, 6.49992505927329333857395184343, 8.011832969610721686895020035599, 8.023034081311283559976527256533, 8.524079546582096676065943493512, 8.965474064429713311069674592028, 9.542604600630706787025977559837
