L(s) = 1 | − 2·5-s + 5·7-s − 3·11-s + 12·13-s − 4·17-s − 4·19-s + 4·23-s + 5·25-s + 10·29-s − 7·31-s − 10·35-s − 4·41-s + 16·43-s − 2·47-s + 18·49-s + 10·53-s + 6·55-s − 9·59-s + 8·61-s − 24·65-s + 6·67-s + 24·71-s + 11·73-s − 15·77-s − 79-s + 30·83-s + 8·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.88·7-s − 0.904·11-s + 3.32·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 25-s + 1.85·29-s − 1.25·31-s − 1.69·35-s − 0.624·41-s + 2.43·43-s − 0.291·47-s + 18/7·49-s + 1.37·53-s + 0.809·55-s − 1.17·59-s + 1.02·61-s − 2.97·65-s + 0.733·67-s + 2.84·71-s + 1.28·73-s − 1.70·77-s − 0.112·79-s + 3.29·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.205958361\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.205958361\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−9.376694052532446034408918411266, −9.177803805478452960543132314568, −8.597061421258669667222882239348, −8.400638206995269276785213430887, −8.160354777585803268312950093429, −8.032677717087618361400923397047, −7.15234666870312418920262191525, −7.04484795662703222322487643593, −6.26793757073321076773438340087, −6.18958289382464225742915066893, −5.36511006869291827397666501949, −5.16364083859426110812322377308, −4.54989270091038728997487255339, −4.24908049407290959179895575174, −3.64573618353519178857988394556, −3.49481735598403349573924349875, −2.39594649831669469681774526345, −2.17396195477479501661724827651, −1.10755901281562975072652870162, −0.915592310022598177256712504551,
0.915592310022598177256712504551, 1.10755901281562975072652870162, 2.17396195477479501661724827651, 2.39594649831669469681774526345, 3.49481735598403349573924349875, 3.64573618353519178857988394556, 4.24908049407290959179895575174, 4.54989270091038728997487255339, 5.16364083859426110812322377308, 5.36511006869291827397666501949, 6.18958289382464225742915066893, 6.26793757073321076773438340087, 7.04484795662703222322487643593, 7.15234666870312418920262191525, 8.032677717087618361400923397047, 8.160354777585803268312950093429, 8.400638206995269276785213430887, 8.597061421258669667222882239348, 9.177803805478452960543132314568, 9.376694052532446034408918411266