| L(s) = 1 | − 4·5-s − 2·9-s + 4·13-s + 2·17-s + 2·25-s − 20·29-s + 12·37-s − 12·41-s + 8·45-s − 10·49-s − 20·53-s + 28·61-s − 16·65-s − 28·73-s − 5·81-s − 8·85-s − 20·89-s + 4·97-s + 20·101-s + 12·109-s + 20·113-s − 8·117-s + 14·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2/3·9-s + 1.10·13-s + 0.485·17-s + 2/5·25-s − 3.71·29-s + 1.97·37-s − 1.87·41-s + 1.19·45-s − 1.42·49-s − 2.74·53-s + 3.58·61-s − 1.98·65-s − 3.27·73-s − 5/9·81-s − 0.867·85-s − 2.11·89-s + 0.406·97-s + 1.99·101-s + 1.14·109-s + 1.88·113-s − 0.739·117-s + 1.27·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.867199747238314258507049095404, −9.721938483460284802573704470251, −8.878376583314565709515988150650, −8.306702051055435143424050976414, −8.119321761242708900614562817168, −7.35622902308931120005953550776, −7.22959647173292270161762928560, −5.94766297465593937866721260525, −5.94691251018865401690641595052, −4.93933220983461313213877514067, −4.14991391129811641343695020033, −3.59025133227290315875645783201, −3.24743736182506202340620349352, −1.78722102085145557690346531679, 0,
1.78722102085145557690346531679, 3.24743736182506202340620349352, 3.59025133227290315875645783201, 4.14991391129811641343695020033, 4.93933220983461313213877514067, 5.94691251018865401690641595052, 5.94766297465593937866721260525, 7.22959647173292270161762928560, 7.35622902308931120005953550776, 8.119321761242708900614562817168, 8.306702051055435143424050976414, 8.878376583314565709515988150650, 9.721938483460284802573704470251, 9.867199747238314258507049095404
