| L(s) = 1 | − 2·3-s + 7-s + 9-s − 4·19-s − 2·21-s − 6·25-s + 4·27-s − 8·31-s + 12·37-s + 49-s + 8·57-s + 63-s + 12·75-s − 11·81-s + 16·93-s + 24·103-s + 20·109-s − 24·111-s + 22·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.917·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s − 1.43·31-s + 1.97·37-s + 1/7·49-s + 1.05·57-s + 0.125·63-s + 1.38·75-s − 1.22·81-s + 1.65·93-s + 2.36·103-s + 1.91·109-s − 2.27·111-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-s + 0.0854·137-s + 0.0848·139-s − 0.164·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.83460272600679831633866855654, −7.64773642258792809430236308287, −7.12993939344099868872995516313, −6.54917291029764894707310980399, −6.11375669138658438283354282507, −5.83478159144884955874857381724, −5.42356324438952837716394427664, −4.73811067375666119619475458270, −4.50618065946357639561135059990, −3.86173562395164512693688006609, −3.31506639401361223413534799422, −2.41829280071188479020496819536, −1.93998681675774983626168415103, −0.983164125137457375820266105317, 0,
0.983164125137457375820266105317, 1.93998681675774983626168415103, 2.41829280071188479020496819536, 3.31506639401361223413534799422, 3.86173562395164512693688006609, 4.50618065946357639561135059990, 4.73811067375666119619475458270, 5.42356324438952837716394427664, 5.83478159144884955874857381724, 6.11375669138658438283354282507, 6.54917291029764894707310980399, 7.12993939344099868872995516313, 7.64773642258792809430236308287, 7.83460272600679831633866855654
