| L(s) = 1 | + 4·5-s − 9-s − 4·11-s + 4·19-s + 11·25-s + 4·29-s + 8·31-s − 12·41-s − 4·45-s − 2·49-s − 16·55-s − 28·59-s + 20·61-s − 16·71-s − 16·79-s + 81-s + 36·89-s + 16·95-s + 4·99-s − 28·101-s − 24·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 1.20·11-s + 0.917·19-s + 11/5·25-s + 0.742·29-s + 1.43·31-s − 1.87·41-s − 0.596·45-s − 2/7·49-s − 2.15·55-s − 3.64·59-s + 2.56·61-s − 1.89·71-s − 1.80·79-s + 1/9·81-s + 3.81·89-s + 1.64·95-s + 0.402·99-s − 2.78·101-s − 2.29·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.020388928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.020388928\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−9.158093684458620614520832841026, −8.483156927793049768175057895261, −8.203396142097353702658192561100, −7.73684288564244224274223851958, −7.51461354384254419417809268475, −6.72837551171062963927816350481, −6.62548920817660021423464808585, −6.31509713445386782909366190284, −5.72211124218642380475171809840, −5.39054375769800099634186200050, −5.29036757156116844128161869765, −4.58438088874882496699477259999, −4.56153144322455286771614073684, −3.59120485750435785431552244293, −3.05962880074096077920680276795, −2.75464441049776245883667430030, −2.49223479740000434515089261364, −1.63632091923718299215314129535, −1.46243885141435954861040462657, −0.52558357502861835195655003683,
0.52558357502861835195655003683, 1.46243885141435954861040462657, 1.63632091923718299215314129535, 2.49223479740000434515089261364, 2.75464441049776245883667430030, 3.05962880074096077920680276795, 3.59120485750435785431552244293, 4.56153144322455286771614073684, 4.58438088874882496699477259999, 5.29036757156116844128161869765, 5.39054375769800099634186200050, 5.72211124218642380475171809840, 6.31509713445386782909366190284, 6.62548920817660021423464808585, 6.72837551171062963927816350481, 7.51461354384254419417809268475, 7.73684288564244224274223851958, 8.203396142097353702658192561100, 8.483156927793049768175057895261, 9.158093684458620614520832841026
