L(s) = 1 | − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 7-s − 8-s + 6·9-s + 3·10-s − 3·12-s − 2·13-s + 14-s + 9·15-s + 16-s − 5·17-s − 6·18-s − 5·19-s − 3·20-s + 3·21-s − 8·23-s + 3·24-s + 4·25-s + 2·26-s − 9·27-s − 28-s + 4·29-s − 9·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 0.866·12-s − 0.554·13-s + 0.267·14-s + 2.32·15-s + 1/4·16-s − 1.21·17-s − 1.41·18-s − 1.14·19-s − 0.670·20-s + 0.654·21-s − 1.66·23-s + 0.612·24-s + 4/5·25-s + 0.392·26-s − 1.73·27-s − 0.188·28-s + 0.742·29-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T - 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−14.0088665027, −13.3628348839, −12.7737890250, −12.4539273437, −12.2393640173, −11.7207970787, −11.4012893253, −11.0323451915, −10.6475347726, −10.1903528358, −9.82917301153, −9.27478100975, −8.60353727838, −8.06242082456, −7.93127524408, −6.98731957165, −6.80451348156, −6.43467492518, −5.90337122031, −5.07814562508, −4.74933509868, −4.03882601977, −3.65962170062, −2.50611097215, −1.60110997374, 0, 0,
1.60110997374, 2.50611097215, 3.65962170062, 4.03882601977, 4.74933509868, 5.07814562508, 5.90337122031, 6.43467492518, 6.80451348156, 6.98731957165, 7.93127524408, 8.06242082456, 8.60353727838, 9.27478100975, 9.82917301153, 10.1903528358, 10.6475347726, 11.0323451915, 11.4012893253, 11.7207970787, 12.2393640173, 12.4539273437, 12.7737890250, 13.3628348839, 14.0088665027