| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s − 4·10-s + 2·11-s − 6·12-s − 8·13-s + 2·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s + 8·19-s + 6·20-s + 2·21-s − 4·22-s − 9·23-s + 8·24-s + 3·25-s + 16·26-s − 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s − 1.73·12-s − 2.21·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 1.83·19-s + 1.34·20-s + 0.436·21-s − 0.852·22-s − 1.87·23-s + 1.63·24-s + 3/5·25-s + 3.13·26-s − 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9049055749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9049055749\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15 T + 134 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 154 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.058304830513310511418852477823, −8.044336531667542665213937731954, −7.41478031892098598841737577565, −7.35483143834538702932315332468, −7.02548105609422570947197930081, −6.62797740994964576833693780278, −6.20850301370970285413606459607, −5.79943890746369625948709625261, −5.51266963798826842399347047975, −5.46658654340193907174344207488, −4.63164546312937028143441318362, −4.58430035306683513395586373239, −3.69746589021308049170562338408, −3.49237602871960727439293704054, −2.72374669469739254582712587344, −2.45410394658772014578618962496, −1.73917000213990662085948766465, −1.71493909730738724934212066852, −0.71719443631182514220201277305, −0.49688705346073905235472774821,
0.49688705346073905235472774821, 0.71719443631182514220201277305, 1.71493909730738724934212066852, 1.73917000213990662085948766465, 2.45410394658772014578618962496, 2.72374669469739254582712587344, 3.49237602871960727439293704054, 3.69746589021308049170562338408, 4.58430035306683513395586373239, 4.63164546312937028143441318362, 5.46658654340193907174344207488, 5.51266963798826842399347047975, 5.79943890746369625948709625261, 6.20850301370970285413606459607, 6.62797740994964576833693780278, 7.02548105609422570947197930081, 7.35483143834538702932315332468, 7.41478031892098598841737577565, 8.044336531667542665213937731954, 8.058304830513310511418852477823
