| L(s) = 1 | − 3·3-s + 5-s + 3·9-s + 5·11-s − 6·13-s − 3·15-s + 17-s − 6·19-s − 6·23-s − 18·29-s + 4·31-s − 15·33-s − 2·37-s + 18·39-s − 8·41-s + 20·43-s + 3·45-s + 47-s − 3·51-s − 4·53-s + 5·55-s + 18·57-s + 8·59-s + 8·61-s − 6·65-s − 12·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 9-s + 1.50·11-s − 1.66·13-s − 0.774·15-s + 0.242·17-s − 1.37·19-s − 1.25·23-s − 3.34·29-s + 0.718·31-s − 2.61·33-s − 0.328·37-s + 2.88·39-s − 1.24·41-s + 3.04·43-s + 0.447·45-s + 0.145·47-s − 0.420·51-s − 0.549·53-s + 0.674·55-s + 2.38·57-s + 1.04·59-s + 1.02·61-s − 0.744·65-s − 1.46·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3456594910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3456594910\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−10.10029021743968538023671421925, −9.934430994249639686033959697071, −9.374713437772091599844849458417, −9.267628747341843134584118262387, −8.563787333573754041189024056033, −8.140181850176752005969610980992, −7.44993758185122295633808126641, −7.13796872600580535725060506007, −6.79469758201643593411032712868, −6.14958816115782488649017942668, −5.91614061787608399566901158135, −5.58222151954653710203048705420, −5.19033671783751511397496297411, −4.52325463981775265974686942944, −3.99935389507306078874350177370, −3.81359052049937885899183199778, −2.64709451011125062993507827783, −2.09494057477336641200053054371, −1.48724060963417655401236893420, −0.30127134837684575171694354925,
0.30127134837684575171694354925, 1.48724060963417655401236893420, 2.09494057477336641200053054371, 2.64709451011125062993507827783, 3.81359052049937885899183199778, 3.99935389507306078874350177370, 4.52325463981775265974686942944, 5.19033671783751511397496297411, 5.58222151954653710203048705420, 5.91614061787608399566901158135, 6.14958816115782488649017942668, 6.79469758201643593411032712868, 7.13796872600580535725060506007, 7.44993758185122295633808126641, 8.140181850176752005969610980992, 8.563787333573754041189024056033, 9.267628747341843134584118262387, 9.374713437772091599844849458417, 9.934430994249639686033959697071, 10.10029021743968538023671421925
