| L(s) = 1 | − 5-s + 5·11-s − 6·17-s − 10·19-s − 6·23-s − 10·29-s − 2·31-s + 8·37-s − 3·41-s + 3·43-s − 4·47-s + 7·49-s + 12·53-s − 5·55-s + 3·59-s − 2·61-s − 11·67-s − 28·71-s − 30·73-s + 10·79-s + 12·83-s + 6·85-s − 28·89-s + 10·95-s + 13·97-s − 12·101-s − 34·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s − 1.45·17-s − 2.29·19-s − 1.25·23-s − 1.85·29-s − 0.359·31-s + 1.31·37-s − 0.468·41-s + 0.457·43-s − 0.583·47-s + 49-s + 1.64·53-s − 0.674·55-s + 0.390·59-s − 0.256·61-s − 1.34·67-s − 3.32·71-s − 3.51·73-s + 1.12·79-s + 1.31·83-s + 0.650·85-s − 2.96·89-s + 1.02·95-s + 1.31·97-s − 1.19·101-s − 3.28·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4142232826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4142232826\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.527254866038541719902160716674, −8.940403965776297884536880497636, −8.655241520453248167470365463302, −8.114780567337784726864762637992, −7.78766142118267506199257065619, −7.14998900217653976309035900039, −7.06717807493524218520804241978, −6.42215607730218376178347788465, −6.28680717671166035417709363644, −5.70733891978350759841606202092, −5.52730855089390466044360912338, −4.51604435548213052849703621204, −4.26989429313363851772515879910, −3.98978339108529072480769504268, −3.91767374355500027577096924479, −2.84072250887378718078314778442, −2.53432958240295813330293143094, −1.69561167442075605649894026703, −1.59168597887098548294120467714, −0.21552206557671963973280089429,
0.21552206557671963973280089429, 1.59168597887098548294120467714, 1.69561167442075605649894026703, 2.53432958240295813330293143094, 2.84072250887378718078314778442, 3.91767374355500027577096924479, 3.98978339108529072480769504268, 4.26989429313363851772515879910, 4.51604435548213052849703621204, 5.52730855089390466044360912338, 5.70733891978350759841606202092, 6.28680717671166035417709363644, 6.42215607730218376178347788465, 7.06717807493524218520804241978, 7.14998900217653976309035900039, 7.78766142118267506199257065619, 8.114780567337784726864762637992, 8.655241520453248167470365463302, 8.940403965776297884536880497636, 9.527254866038541719902160716674
