| L(s) = 1 | + 2·2-s − 16·13-s + 24·17-s + 10·25-s − 32·26-s + 8·29-s − 32·32-s + 48·34-s + 16·37-s + 112·41-s + 96·49-s + 20·50-s + 176·53-s + 16·58-s + 128·61-s − 64·64-s + 264·73-s + 32·74-s + 224·82-s + 88·89-s − 264·97-s + 192·98-s − 328·101-s + 352·106-s − 128·109-s − 504·113-s + 84·121-s + ⋯ |
| L(s) = 1 | + 2-s − 1.23·13-s + 1.41·17-s + 2/5·25-s − 1.23·26-s + 8/29·29-s − 32-s + 1.41·34-s + 0.432·37-s + 2.73·41-s + 1.95·49-s + 2/5·50-s + 3.32·53-s + 8/29·58-s + 2.09·61-s − 64-s + 3.61·73-s + 0.432·74-s + 2.73·82-s + 0.988·89-s − 2.72·97-s + 1.95·98-s − 3.24·101-s + 3.32·106-s − 1.17·109-s − 4.46·113-s + 0.694·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.644232411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.644232411\) |
| \(L(2)\) |
|
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 | $C_4$ | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2^2:C_4$ | \( 1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 334 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 294 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 1606 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 2254 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 56 T + 3966 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 6896 T^{2} + 18665806 T^{4} - 6896 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 88 T + 7054 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 64 T + 5086 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 7536 T^{2} + 47276046 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 132 T + 9894 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 21296 T^{2} + 201120526 T^{4} - 21296 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 44 T + 8326 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.112993583063023573389939760877, −8.810136251039589896746596715265, −8.357881654070276076538388048451, −8.269546099363164904621286182288, −7.79653319208947233388284032214, −7.57315541498131009025042517263, −7.56476722132588954222966461423, −6.90661049569924219153260645798, −6.82252430635011132095208689330, −6.65144728692677667916969098929, −6.17013446208395822952469741981, −5.45372050141977323325365379743, −5.41885683525770648456837903213, −5.34205980062353352062302120108, −5.34113970026599315939787309665, −4.45061087105230734621054161227, −4.18585968978765290348821271261, −3.96119809043752482827936122196, −3.94213392995068983698074401931, −3.18716029263427950265925992322, −2.60865586196564007459598000088, −2.57717710395715256225166291396, −2.05194167619393412273473921435, −1.06184625448812135752651233757, −0.69676874048552710604398229926,
0.69676874048552710604398229926, 1.06184625448812135752651233757, 2.05194167619393412273473921435, 2.57717710395715256225166291396, 2.60865586196564007459598000088, 3.18716029263427950265925992322, 3.94213392995068983698074401931, 3.96119809043752482827936122196, 4.18585968978765290348821271261, 4.45061087105230734621054161227, 5.34113970026599315939787309665, 5.34205980062353352062302120108, 5.41885683525770648456837903213, 5.45372050141977323325365379743, 6.17013446208395822952469741981, 6.65144728692677667916969098929, 6.82252430635011132095208689330, 6.90661049569924219153260645798, 7.56476722132588954222966461423, 7.57315541498131009025042517263, 7.79653319208947233388284032214, 8.269546099363164904621286182288, 8.357881654070276076538388048451, 8.810136251039589896746596715265, 9.112993583063023573389939760877
