| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 2·9-s + 8·10-s − 6·11-s − 4·12-s − 6·13-s − 8·15-s − 4·16-s + 4·18-s − 2·19-s + 8·20-s − 12·22-s + 16·23-s + 11·25-s − 12·26-s − 6·27-s + 6·29-s − 16·30-s − 8·32-s + 12·33-s + 4·36-s − 6·37-s − 4·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 2/3·9-s + 2.52·10-s − 1.80·11-s − 1.15·12-s − 1.66·13-s − 2.06·15-s − 16-s + 0.942·18-s − 0.458·19-s + 1.78·20-s − 2.55·22-s + 3.33·23-s + 11/5·25-s − 2.35·26-s − 1.15·27-s + 1.11·29-s − 2.92·30-s − 1.41·32-s + 2.08·33-s + 2/3·36-s − 0.986·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398411631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398411631\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 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
−14.57742198816264828588736620394, −14.04901381589848517669287105152, −13.21263327872961666426845699339, −13.12580103815786168627567314299, −12.85605969118348066594036764241, −12.17801621362332164398493210095, −11.49524970232935581612553876823, −10.96056544901509384935604857787, −10.26075897927273957609018588390, −10.07059388567639789516193372782, −9.228785443790332110863380008958, −8.550786391460375997190461702687, −7.18837648543051429883700073371, −6.91565115638612820206505696023, −6.13044312346254729524365106867, −5.28560249338780965286128836829, −5.16957075528633276263265883990, −4.81303490109649790741175582599, −3.00055124653600007170672131703, −2.35800388573391443252054775612,
2.35800388573391443252054775612, 3.00055124653600007170672131703, 4.81303490109649790741175582599, 5.16957075528633276263265883990, 5.28560249338780965286128836829, 6.13044312346254729524365106867, 6.91565115638612820206505696023, 7.18837648543051429883700073371, 8.550786391460375997190461702687, 9.228785443790332110863380008958, 10.07059388567639789516193372782, 10.26075897927273957609018588390, 10.96056544901509384935604857787, 11.49524970232935581612553876823, 12.17801621362332164398493210095, 12.85605969118348066594036764241, 13.12580103815786168627567314299, 13.21263327872961666426845699339, 14.04901381589848517669287105152, 14.57742198816264828588736620394
