| L(s) = 1 | + 2·5-s + 6·9-s + 8·19-s − 25-s − 4·29-s + 16·31-s + 12·41-s + 12·45-s − 49-s − 8·59-s − 12·61-s − 24·71-s − 8·79-s + 27·81-s + 20·89-s + 16·95-s − 36·101-s − 28·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2·9-s + 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s + 1.87·41-s + 1.78·45-s − 1/7·49-s − 1.04·59-s − 1.53·61-s − 2.84·71-s − 0.900·79-s + 3·81-s + 2.11·89-s + 1.64·95-s − 3.58·101-s − 2.68·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.867457141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.867457141\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.64621296412020896649484374983, −10.55710139719972886206778780483, −10.06291567971881036302637130340, −9.624704648885627346441251697559, −9.355483783050034522504474973207, −9.134949576564553691895301969857, −8.135948951116062401115164912635, −7.83201703740521485878140838163, −7.38009549533829100373763125701, −7.03747731098302302180039020275, −6.27437252788839500920666443081, −6.14924573008747085200813762455, −5.40777674015997082377076974892, −4.90098488757384188230680961111, −4.32812241137330901760803721151, −4.00101149824062534662943114451, −3.02442353022434026574576407813, −2.57548060170950369116985658849, −1.48518091307567065337490613066, −1.21483707228753499924510782515,
1.21483707228753499924510782515, 1.48518091307567065337490613066, 2.57548060170950369116985658849, 3.02442353022434026574576407813, 4.00101149824062534662943114451, 4.32812241137330901760803721151, 4.90098488757384188230680961111, 5.40777674015997082377076974892, 6.14924573008747085200813762455, 6.27437252788839500920666443081, 7.03747731098302302180039020275, 7.38009549533829100373763125701, 7.83201703740521485878140838163, 8.135948951116062401115164912635, 9.134949576564553691895301969857, 9.355483783050034522504474973207, 9.624704648885627346441251697559, 10.06291567971881036302637130340, 10.55710139719972886206778780483, 10.64621296412020896649484374983
