| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 5-s + 2·6-s + 7-s − 4·8-s − 2·10-s + 3·11-s − 3·12-s − 2·13-s − 2·14-s − 15-s + 5·16-s − 6·17-s − 2·19-s + 3·20-s − 21-s − 6·22-s + 12·23-s + 4·24-s + 4·26-s + 27-s + 3·28-s + 18·29-s + 2·30-s − 7·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s − 1.41·8-s − 0.632·10-s + 0.904·11-s − 0.866·12-s − 0.554·13-s − 0.534·14-s − 0.258·15-s + 5/4·16-s − 1.45·17-s − 0.458·19-s + 0.670·20-s − 0.218·21-s − 1.27·22-s + 2.50·23-s + 0.816·24-s + 0.784·26-s + 0.192·27-s + 0.566·28-s + 3.34·29-s + 0.365·30-s − 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8126413876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8126413876\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−10.30820555418554570971033039596, −9.640044815890584304711187261602, −9.636117515485580174543391905415, −8.821014470046287667059007273344, −8.717754513599062075332712427137, −8.470043355380162917765803633227, −7.910285913194898528289174964703, −7.06196729034940255194392215070, −6.84043576405043844082164386713, −6.74909678893546006042465072379, −6.28211456499897401376617156653, −5.53288856421949049965336521547, −5.03606763210404721564561377676, −4.72185246044900715789996531479, −4.06668902048669258430836161170, −3.06703750625511754457083983035, −2.81436069562022185395438184424, −1.91641237494418434948207602502, −1.43845497735194352975432962682, −0.58192753142310427380954381581,
0.58192753142310427380954381581, 1.43845497735194352975432962682, 1.91641237494418434948207602502, 2.81436069562022185395438184424, 3.06703750625511754457083983035, 4.06668902048669258430836161170, 4.72185246044900715789996531479, 5.03606763210404721564561377676, 5.53288856421949049965336521547, 6.28211456499897401376617156653, 6.74909678893546006042465072379, 6.84043576405043844082164386713, 7.06196729034940255194392215070, 7.910285913194898528289174964703, 8.470043355380162917765803633227, 8.717754513599062075332712427137, 8.821014470046287667059007273344, 9.636117515485580174543391905415, 9.640044815890584304711187261602, 10.30820555418554570971033039596
