| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 8-s + 10-s + 4·11-s − 4·13-s + 15-s − 16-s − 2·17-s − 4·19-s − 4·22-s + 8·23-s − 24-s + 4·26-s + 27-s − 4·29-s − 30-s − 4·33-s + 2·34-s − 6·37-s + 4·38-s + 4·39-s − 40-s − 12·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.10·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.852·22-s + 1.66·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.742·29-s − 0.182·30-s − 0.696·33-s + 0.342·34-s − 0.986·37-s + 0.648·38-s + 0.640·39-s − 0.158·40-s − 1.87·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1503626487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1503626487\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.921786504769293386969992723700, −9.029643436443142395763086182464, −8.926062439579988282733805775942, −8.748141184713575991575708337038, −8.218608412961597337015454904071, −7.65486644414546973039491455412, −7.18815452815690016480526597109, −6.91338794009808942062493779930, −6.72958985104750366958170501717, −6.05657233091380281128283240236, −5.64100714543990837945190713085, −4.96733083547741726724237868845, −4.81701826881290538766737410614, −4.20987434535950625249596678030, −3.84284692218780589480209114128, −3.08443276671761630746816043011, −2.71296431830910297528955304249, −1.60185157000421061617489455652, −1.50866754360926486913824514006, −0.18533586957558062056531736026,
0.18533586957558062056531736026, 1.50866754360926486913824514006, 1.60185157000421061617489455652, 2.71296431830910297528955304249, 3.08443276671761630746816043011, 3.84284692218780589480209114128, 4.20987434535950625249596678030, 4.81701826881290538766737410614, 4.96733083547741726724237868845, 5.64100714543990837945190713085, 6.05657233091380281128283240236, 6.72958985104750366958170501717, 6.91338794009808942062493779930, 7.18815452815690016480526597109, 7.65486644414546973039491455412, 8.218608412961597337015454904071, 8.748141184713575991575708337038, 8.926062439579988282733805775942, 9.029643436443142395763086182464, 9.921786504769293386969992723700
