| L(s) = 1 | − 4-s + 16-s − 8·19-s + 4·29-s − 16·31-s + 12·41-s − 2·49-s − 20·61-s − 64-s + 32·71-s + 8·76-s + 16·79-s − 28·89-s − 20·101-s − 28·109-s − 4·116-s − 22·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 12·164-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 1.83·19-s + 0.742·29-s − 2.87·31-s + 1.87·41-s − 2/7·49-s − 2.56·61-s − 1/8·64-s + 3.79·71-s + 0.917·76-s + 1.80·79-s − 2.96·89-s − 1.99·101-s − 2.68·109-s − 0.371·116-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 0.937·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4581519714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4581519714\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.534152820672841764531475712962, −7.919988073666175394694549378834, −7.76196405449442076734416771721, −7.13918252407822289133307190687, −6.89084006437146102587944905895, −6.37891583083903152321327954100, −6.34512414502128196199245344892, −5.65057238785939541610352624124, −5.34517030754518687892604002487, −5.23686293210021983107872850964, −4.43793212707552996690108082970, −4.28631455254550334423450347841, −3.99349061079054980620793294183, −3.50380977256844395006186230173, −3.08044305662961324279902293412, −2.44541812049859460712219601008, −2.18883706441261015244961545066, −1.60092374720150283550564529370, −1.06092313251401736115090796316, −0.18852261676169983550712432673,
0.18852261676169983550712432673, 1.06092313251401736115090796316, 1.60092374720150283550564529370, 2.18883706441261015244961545066, 2.44541812049859460712219601008, 3.08044305662961324279902293412, 3.50380977256844395006186230173, 3.99349061079054980620793294183, 4.28631455254550334423450347841, 4.43793212707552996690108082970, 5.23686293210021983107872850964, 5.34517030754518687892604002487, 5.65057238785939541610352624124, 6.34512414502128196199245344892, 6.37891583083903152321327954100, 6.89084006437146102587944905895, 7.13918252407822289133307190687, 7.76196405449442076734416771721, 7.919988073666175394694549378834, 8.534152820672841764531475712962
