| L(s) = 1 | − 3-s − 4·4-s − 2·9-s + 4·12-s + 12·16-s − 25-s + 5·27-s − 10·31-s + 8·36-s + 14·37-s − 12·48-s − 14·49-s − 32·64-s + 26·67-s + 75-s + 81-s + 10·93-s + 34·97-s + 4·100-s − 8·103-s − 20·108-s − 14·111-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 2/3·9-s + 1.15·12-s + 3·16-s − 1/5·25-s + 0.962·27-s − 1.79·31-s + 4/3·36-s + 2.30·37-s − 1.73·48-s − 2·49-s − 4·64-s + 3.17·67-s + 0.115·75-s + 1/9·81-s + 1.03·93-s + 3.45·97-s + 2/5·100-s − 0.788·103-s − 1.92·108-s − 1.32·111-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−9.239645513051175281393608873936, −8.487357851440445247270968425966, −8.468596719955261371677845250157, −7.74971489374349908553225834156, −7.37520405553462112512317968814, −6.34583551331242224592183459087, −6.04214609726225978583223790134, −5.46569305050405475214160980267, −4.87628538913627800967773832068, −4.73070320716804110076258258958, −3.66923064750984294129055958828, −3.60577326155639270878357097593, −2.44495639562721598403091577465, −1.06294681607902093882729364169, 0,
1.06294681607902093882729364169, 2.44495639562721598403091577465, 3.60577326155639270878357097593, 3.66923064750984294129055958828, 4.73070320716804110076258258958, 4.87628538913627800967773832068, 5.46569305050405475214160980267, 6.04214609726225978583223790134, 6.34583551331242224592183459087, 7.37520405553462112512317968814, 7.74971489374349908553225834156, 8.468596719955261371677845250157, 8.487357851440445247270968425966, 9.239645513051175281393608873936
