| L(s) = 1 | − 2·3-s − 4-s − 4·7-s + 9-s − 6·11-s + 2·12-s + 16-s + 8·21-s + 25-s + 4·27-s + 4·28-s + 12·33-s − 36-s − 7·37-s + 6·44-s − 12·47-s − 2·48-s − 2·49-s − 4·63-s − 64-s − 22·67-s − 6·71-s + 5·73-s − 2·75-s + 24·77-s − 11·81-s − 18·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 1/4·16-s + 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.755·28-s + 2.08·33-s − 1/6·36-s − 1.15·37-s + 0.904·44-s − 1.75·47-s − 0.288·48-s − 2/7·49-s − 0.503·63-s − 1/8·64-s − 2.68·67-s − 0.712·71-s + 0.585·73-s − 0.230·75-s + 2.73·77-s − 1.22·81-s − 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−8.101089558721439876676913392373, −7.68908270873636649226067324224, −7.03831401760548714683662404258, −6.72218287273097182072480181807, −6.11962015880872700762066401901, −5.86004128903805709268947809887, −5.24023520403489755086373703147, −4.97181410840635772342179465724, −4.42941561727296000640350556826, −3.60494361823484161472828088410, −3.02784421438766372236027129404, −2.69329772189143910717835006499, −1.48012435535376598886585370530, 0, 0,
1.48012435535376598886585370530, 2.69329772189143910717835006499, 3.02784421438766372236027129404, 3.60494361823484161472828088410, 4.42941561727296000640350556826, 4.97181410840635772342179465724, 5.24023520403489755086373703147, 5.86004128903805709268947809887, 6.11962015880872700762066401901, 6.72218287273097182072480181807, 7.03831401760548714683662404258, 7.68908270873636649226067324224, 8.101089558721439876676913392373
