| L(s) = 1 | − 2·2-s + 3·5-s + 7-s + 4·8-s − 6·10-s − 5·11-s − 13-s − 2·14-s − 4·16-s + 3·17-s − 9·19-s + 10·22-s − 3·23-s + 2·25-s + 2·26-s + 5·29-s − 31-s − 6·34-s + 3·35-s + 18·38-s + 12·40-s + 5·41-s + 6·43-s + 6·46-s − 7·49-s − 4·50-s − 53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.34·5-s + 0.377·7-s + 1.41·8-s − 1.89·10-s − 1.50·11-s − 0.277·13-s − 0.534·14-s − 16-s + 0.727·17-s − 2.06·19-s + 2.13·22-s − 0.625·23-s + 2/5·25-s + 0.392·26-s + 0.928·29-s − 0.179·31-s − 1.02·34-s + 0.507·35-s + 2.91·38-s + 1.89·40-s + 0.780·41-s + 0.914·43-s + 0.884·46-s − 49-s − 0.565·50-s − 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2885064565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2885064565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 50 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 61 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 106 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 135 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−19.5082415633, −19.0234623439, −18.6347746586, −17.9982729700, −17.7078996587, −17.3107248098, −16.8557025546, −16.1093797206, −15.4802633775, −14.4158764913, −14.2693216873, −13.3232931067, −13.0112683442, −12.3279847791, −11.0743273627, −10.4507701525, −10.0283749808, −9.58517150303, −8.68182783245, −8.23349910033, −7.56845968617, −6.29572249493, −5.42702336844, −4.48303726399, −2.25941107232,
2.25941107232, 4.48303726399, 5.42702336844, 6.29572249493, 7.56845968617, 8.23349910033, 8.68182783245, 9.58517150303, 10.0283749808, 10.4507701525, 11.0743273627, 12.3279847791, 13.0112683442, 13.3232931067, 14.2693216873, 14.4158764913, 15.4802633775, 16.1093797206, 16.8557025546, 17.3107248098, 17.7078996587, 17.9982729700, 18.6347746586, 19.0234623439, 19.5082415633
