| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·10-s − 4·16-s + 2·20-s + 25-s + 4·31-s + 8·32-s + 2·49-s − 2·50-s − 20·53-s − 8·62-s − 8·64-s − 4·79-s − 4·80-s − 4·98-s + 2·100-s + 40·106-s − 40·107-s − 18·121-s + 8·124-s + 125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s − 16-s + 0.447·20-s + 1/5·25-s + 0.718·31-s + 1.41·32-s + 2/7·49-s − 0.282·50-s − 2.74·53-s − 1.01·62-s − 64-s − 0.450·79-s − 0.447·80-s − 0.404·98-s + 1/5·100-s + 3.88·106-s − 3.86·107-s − 1.63·121-s + 0.718·124-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−8.153543152034258994925509043730, −7.933732008215809077651917197263, −7.32883362927995997264935678755, −6.94342607708819905831692540985, −6.38568810670993783128133879071, −6.15663579835648770188191046984, −5.38264816632881500976502432582, −4.93845168095510009537710950835, −4.39964399140920609147943599144, −3.80943787946265873192775063583, −2.97072343093666435208355350892, −2.50426452219507246887557404802, −1.69785640981608975573764053348, −1.16491078036491943327884266774, 0,
1.16491078036491943327884266774, 1.69785640981608975573764053348, 2.50426452219507246887557404802, 2.97072343093666435208355350892, 3.80943787946265873192775063583, 4.39964399140920609147943599144, 4.93845168095510009537710950835, 5.38264816632881500976502432582, 6.15663579835648770188191046984, 6.38568810670993783128133879071, 6.94342607708819905831692540985, 7.32883362927995997264935678755, 7.933732008215809077651917197263, 8.153543152034258994925509043730
