| L(s) = 1 | + 2·9-s + 2·11-s − 8·19-s + 12·29-s − 16·31-s + 12·41-s − 2·49-s − 24·59-s + 4·61-s + 24·71-s + 16·79-s − 5·81-s − 12·89-s + 4·99-s − 36·101-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.603·11-s − 1.83·19-s + 2.22·29-s − 2.87·31-s + 1.87·41-s − 2/7·49-s − 3.12·59-s + 0.512·61-s + 2.84·71-s + 1.80·79-s − 5/9·81-s − 1.27·89-s + 0.402·99-s − 3.58·101-s + 1.91·109-s + 3/11·121-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 + 0.0773·167-s + 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.947566287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947566287\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.484868423424914574257910180988, −8.185068036558618712086829716703, −7.81789478883457532979644268571, −7.53299544089796342107674187294, −6.88503091097514978187677130889, −6.79029853021330597213138029175, −6.48592897144881120962580787079, −5.91397632472205077913348983630, −5.80367757595412118719316053403, −5.08452512656498444500279709867, −4.81270531717960688091592799991, −4.36108758406222173207937981646, −4.00343639222457504145259479806, −3.74676459068602669907655443151, −3.16827334640832864537205762487, −2.59012011811437673221877864792, −2.20733581041761689761448256324, −1.65155429115363665808677536462, −1.21091663369361519384269238451, −0.40377772005865333725877731337,
0.40377772005865333725877731337, 1.21091663369361519384269238451, 1.65155429115363665808677536462, 2.20733581041761689761448256324, 2.59012011811437673221877864792, 3.16827334640832864537205762487, 3.74676459068602669907655443151, 4.00343639222457504145259479806, 4.36108758406222173207937981646, 4.81270531717960688091592799991, 5.08452512656498444500279709867, 5.80367757595412118719316053403, 5.91397632472205077913348983630, 6.48592897144881120962580787079, 6.79029853021330597213138029175, 6.88503091097514978187677130889, 7.53299544089796342107674187294, 7.81789478883457532979644268571, 8.185068036558618712086829716703, 8.484868423424914574257910180988
