| L(s) = 1 | − 3-s − 4·5-s − 9-s + 2·11-s − 5·13-s + 4·15-s + 2·17-s + 2·19-s + 2·23-s + 2·25-s − 3·29-s + 9·31-s − 2·33-s + 5·39-s + 41-s − 16·43-s + 4·45-s − 11·47-s − 14·49-s − 2·51-s − 4·53-s − 8·55-s − 2·57-s + 4·59-s − 8·61-s + 20·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 1/3·9-s + 0.603·11-s − 1.38·13-s + 1.03·15-s + 0.485·17-s + 0.458·19-s + 0.417·23-s + 2/5·25-s − 0.557·29-s + 1.61·31-s − 0.348·33-s + 0.800·39-s + 0.156·41-s − 2.43·43-s + 0.596·45-s − 1.60·47-s − 2·49-s − 0.280·51-s − 0.549·53-s − 1.07·55-s − 0.264·57-s + 0.520·59-s − 1.02·61-s + 2.48·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 23 T + 270 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.382849150871586848509360896520, −8.861593580565456522259430779967, −8.224308489197493315999961099026, −8.132115689447926497710853094440, −7.76958308779848159490365071384, −7.30057266574652109311325660901, −6.92761160832395527081247233340, −6.59242096916300090184090663137, −5.91532066573763485954839514580, −5.69423863866812760076284712130, −4.85083547918706198605540124263, −4.71150820830754754565370046171, −4.39411812799979121380234921574, −3.63732807512071240120920298130, −3.12830673534980721202799260606, −3.06055617911215464314095458708, −1.93430990891018862732677174646, −1.28794341500950429886473213746, 0, 0,
1.28794341500950429886473213746, 1.93430990891018862732677174646, 3.06055617911215464314095458708, 3.12830673534980721202799260606, 3.63732807512071240120920298130, 4.39411812799979121380234921574, 4.71150820830754754565370046171, 4.85083547918706198605540124263, 5.69423863866812760076284712130, 5.91532066573763485954839514580, 6.59242096916300090184090663137, 6.92761160832395527081247233340, 7.30057266574652109311325660901, 7.76958308779848159490365071384, 8.132115689447926497710853094440, 8.224308489197493315999961099026, 8.861593580565456522259430779967, 9.382849150871586848509360896520
