| L(s) = 1 | + 2·2-s + 4-s + 3·5-s − 2·7-s − 4·9-s + 6·10-s + 5·11-s − 3·13-s − 4·14-s + 16-s + 6·17-s − 8·18-s + 5·19-s + 3·20-s + 10·22-s + 3·23-s + 2·25-s − 6·26-s − 2·28-s + 5·29-s − 7·31-s − 2·32-s + 12·34-s − 6·35-s − 4·36-s + 37-s + 10·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 4/3·9-s + 1.89·10-s + 1.50·11-s − 0.832·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 1.88·18-s + 1.14·19-s + 0.670·20-s + 2.13·22-s + 0.625·23-s + 2/5·25-s − 1.17·26-s − 0.377·28-s + 0.928·29-s − 1.25·31-s − 0.353·32-s + 2.05·34-s − 1.01·35-s − 2/3·36-s + 0.164·37-s + 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70771 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70771 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.212444033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.212444033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 181 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 88 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T - 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_4$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 195 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 58 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−14.3457173583, −14.0360061964, −13.6995541740, −13.1050735673, −12.7057588261, −12.3078329129, −11.9958219812, −11.3157784187, −10.9960757745, −10.2185897361, −9.57788619839, −9.49196643265, −9.15477334895, −8.32940848334, −7.65730637155, −7.08860006739, −6.31773834331, −6.08305029326, −5.43174454354, −5.22259287310, −4.50806452305, −3.56466666355, −3.27835936245, −2.51490690242, −1.34740788631,
1.34740788631, 2.51490690242, 3.27835936245, 3.56466666355, 4.50806452305, 5.22259287310, 5.43174454354, 6.08305029326, 6.31773834331, 7.08860006739, 7.65730637155, 8.32940848334, 9.15477334895, 9.49196643265, 9.57788619839, 10.2185897361, 10.9960757745, 11.3157784187, 11.9958219812, 12.3078329129, 12.7057588261, 13.1050735673, 13.6995541740, 14.0360061964, 14.3457173583
