| L(s) = 1 | + 6·5-s − 14·7-s − 20·11-s + 18·13-s − 2·17-s + 46·23-s + 11·25-s − 28·31-s − 84·35-s + 66·37-s + 28·41-s − 30·43-s + 78·47-s + 98·49-s + 14·53-s − 120·55-s + 84·61-s + 108·65-s − 14·67-s − 196·71-s + 98·73-s + 280·77-s + 126·83-s − 12·85-s − 252·91-s + 66·97-s − 52·101-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 2·7-s − 1.81·11-s + 1.38·13-s − 0.117·17-s + 2·23-s + 0.439·25-s − 0.903·31-s − 2.39·35-s + 1.78·37-s + 0.682·41-s − 0.697·43-s + 1.65·47-s + 2·49-s + 0.264·53-s − 2.18·55-s + 1.37·61-s + 1.66·65-s − 0.208·67-s − 2.76·71-s + 1.34·73-s + 3.63·77-s + 1.51·83-s − 0.141·85-s − 2.76·91-s + 0.680·97-s − 0.514·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.635152300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635152300\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1618 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T + 3042 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3826 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3298 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.87611320147649623690295275316, −12.56192552109916513188340971956, −11.63957745364878619939023721189, −10.91943890239847995863245450906, −10.67637820374118812343922874781, −10.20579789019748764004745778801, −9.655469320217962439390715272160, −9.226251452394690047860928128234, −8.885453364659721127959967203300, −8.136288116033429805603106731066, −7.38334890744579500915392245086, −6.87359774180022938643796571344, −6.26437850337862027166954598565, −5.74148252498489561686975386972, −5.48051454771260795604611242705, −4.49482524437903687438284595150, −3.46434196438576122962812653957, −2.94018654793248844038244452645, −2.28260129572071880616325647275, −0.76678388220796601964567880951,
0.76678388220796601964567880951, 2.28260129572071880616325647275, 2.94018654793248844038244452645, 3.46434196438576122962812653957, 4.49482524437903687438284595150, 5.48051454771260795604611242705, 5.74148252498489561686975386972, 6.26437850337862027166954598565, 6.87359774180022938643796571344, 7.38334890744579500915392245086, 8.136288116033429805603106731066, 8.885453364659721127959967203300, 9.226251452394690047860928128234, 9.655469320217962439390715272160, 10.20579789019748764004745778801, 10.67637820374118812343922874781, 10.91943890239847995863245450906, 11.63957745364878619939023721189, 12.56192552109916513188340971956, 12.87611320147649623690295275316
