| L(s) = 1 | − 2-s − 3·3-s + 2·4-s − 5-s + 3·6-s − 5·8-s + 6·9-s + 10-s − 5·11-s − 6·12-s − 5·13-s + 3·15-s + 5·16-s − 6·17-s − 6·18-s − 2·19-s − 2·20-s + 5·22-s − 3·23-s + 15·24-s + 5·25-s + 5·26-s − 9·27-s + 29-s − 3·30-s − 10·32-s + 15·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 4-s − 0.447·5-s + 1.22·6-s − 1.76·8-s + 2·9-s + 0.316·10-s − 1.50·11-s − 1.73·12-s − 1.38·13-s + 0.774·15-s + 5/4·16-s − 1.45·17-s − 1.41·18-s − 0.458·19-s − 0.447·20-s + 1.06·22-s − 0.625·23-s + 3.06·24-s + 25-s + 0.980·26-s − 1.73·27-s + 0.185·29-s − 0.547·30-s − 1.76·32-s + 2.61·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79986000589164937841870418537, −10.70371277507289663796721055705, −10.04680456843492734903442569177, −9.840309209941096572813486996248, −9.015134875689948125772806480985, −8.812356764534351378782297816606, −7.972592848908292992146851702832, −7.57778244473326964945080354131, −7.20582544937681542002073887943, −6.59997082647348295533167961426, −6.24708362972419236706789795104, −5.87729487185656073521172061780, −5.13918503241466453936853480635, −4.74318850001069861386297510355, −4.27111317857515609001967569529, −2.98802852032987997401209236625, −2.65308002061121635953759547720, −1.71499801023174085083386906158, 0, 0,
1.71499801023174085083386906158, 2.65308002061121635953759547720, 2.98802852032987997401209236625, 4.27111317857515609001967569529, 4.74318850001069861386297510355, 5.13918503241466453936853480635, 5.87729487185656073521172061780, 6.24708362972419236706789795104, 6.59997082647348295533167961426, 7.20582544937681542002073887943, 7.57778244473326964945080354131, 7.972592848908292992146851702832, 8.812356764534351378782297816606, 9.015134875689948125772806480985, 9.840309209941096572813486996248, 10.04680456843492734903442569177, 10.70371277507289663796721055705, 10.79986000589164937841870418537
