| L(s) = 1 | − 2·4-s + 2·7-s − 2·9-s − 5·11-s − 3·13-s + 4·16-s + 7·17-s − 6·19-s + 3·23-s + 2·25-s − 4·28-s + 6·29-s − 9·31-s + 4·36-s − 2·37-s + 3·41-s + 7·43-s + 10·44-s + 4·47-s + 2·49-s + 6·52-s + 53-s + 8·59-s − 6·61-s − 4·63-s − 8·64-s + 5·67-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s − 2/3·9-s − 1.50·11-s − 0.832·13-s + 16-s + 1.69·17-s − 1.37·19-s + 0.625·23-s + 2/5·25-s − 0.755·28-s + 1.11·29-s − 1.61·31-s + 2/3·36-s − 0.328·37-s + 0.468·41-s + 1.06·43-s + 1.50·44-s + 0.583·47-s + 2/7·49-s + 0.832·52-s + 0.137·53-s + 1.04·59-s − 0.768·61-s − 0.503·63-s − 64-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4878579029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4878579029\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 28 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 5 T - 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−19.3672709522, −18.7537720085, −18.4626278662, −17.8338835405, −17.2083782326, −16.9779399105, −16.2597328002, −15.4430012049, −14.7338840776, −14.4252980222, −13.9882037578, −13.0593824784, −12.6188422693, −12.1751924739, −11.1065355877, −10.5717750614, −10.0056947939, −9.14502487721, −8.38568144207, −7.95327274223, −7.16610035310, −5.58503558477, −5.30680156586, −4.28617911363, −2.82479881802,
2.82479881802, 4.28617911363, 5.30680156586, 5.58503558477, 7.16610035310, 7.95327274223, 8.38568144207, 9.14502487721, 10.0056947939, 10.5717750614, 11.1065355877, 12.1751924739, 12.6188422693, 13.0593824784, 13.9882037578, 14.4252980222, 14.7338840776, 15.4430012049, 16.2597328002, 16.9779399105, 17.2083782326, 17.8338835405, 18.4626278662, 18.7537720085, 19.3672709522
