| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420211874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420211874\) |
| \(L(\frac{3}{2})\) |
|
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
−13.19936138384739914382352658115, −12.96669995256079151075404019677, −12.56486098484899281556676128666, −12.18572119518809602416269541520, −11.10031189249695439529331808755, −10.79324829091812215006432572837, −10.45585803066577920268658596337, −9.732895421012609469365159122906, −9.187506760102842660251056287206, −9.172468262081889166146317305118, −7.85347736771192485577263380643, −7.73105409135580991251498714852, −6.70175068557014595121269523456, −6.62707383899723269380668387610, −5.65226771146563682921522801148, −5.12471343564457321137671406627, −4.03744434739196994906888648472, −3.93490860985937793236879916016, −2.27423480707383389006180586212, −1.67830909819770971279465816738,
1.67830909819770971279465816738, 2.27423480707383389006180586212, 3.93490860985937793236879916016, 4.03744434739196994906888648472, 5.12471343564457321137671406627, 5.65226771146563682921522801148, 6.62707383899723269380668387610, 6.70175068557014595121269523456, 7.73105409135580991251498714852, 7.85347736771192485577263380643, 9.172468262081889166146317305118, 9.187506760102842660251056287206, 9.732895421012609469365159122906, 10.45585803066577920268658596337, 10.79324829091812215006432572837, 11.10031189249695439529331808755, 12.18572119518809602416269541520, 12.56486098484899281556676128666, 12.96669995256079151075404019677, 13.19936138384739914382352658115
