| L(s) = 1 | + 2-s + 8·4-s + 5·5-s + 6·7-s + 23·8-s + 5·10-s − 47·11-s + 5·13-s + 6·14-s + 23·16-s + 262·17-s − 112·19-s + 40·20-s − 47·22-s + 3·23-s + 5·26-s + 48·28-s − 157·29-s − 225·31-s + 184·32-s + 262·34-s + 30·35-s − 140·37-s − 112·38-s + 115·40-s + 140·41-s − 397·43-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 4-s + 0.447·5-s + 0.323·7-s + 1.01·8-s + 0.158·10-s − 1.28·11-s + 0.106·13-s + 0.114·14-s + 0.359·16-s + 3.73·17-s − 1.35·19-s + 0.447·20-s − 0.455·22-s + 0.0271·23-s + 0.0377·26-s + 0.323·28-s − 1.00·29-s − 1.30·31-s + 1.01·32-s + 1.32·34-s + 0.144·35-s − 0.622·37-s − 0.478·38-s + 0.454·40-s + 0.533·41-s − 1.40·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.162716458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.162716458\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T - 307 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 47 T + 878 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T - 2172 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 131 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 12158 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 157 T + 260 T^{2} + 157 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 225 T + 20834 T^{2} + 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 140 T - 49321 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 397 T + 78102 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 347 T + 16586 T^{2} + 347 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 748 T + 354125 T^{2} - 748 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 338 T - 112737 T^{2} - 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 492 T - 58699 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 970 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1257 T + 1087010 T^{2} - 1257 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 102 T - 561383 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1488 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 974 T + 36003 T^{2} + 974 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.00305315537488317143596368968, −10.54037367682569579920004366535, −10.31714912434841002131519526681, −9.911162063304479401087819879755, −9.396002806365963110901379082504, −8.688742893077145323964999997375, −7.87806314303922216421800441235, −7.86450514138027266808410492180, −7.54490658990155895428086736956, −6.70950385483033090187267650416, −6.41626014943428114089570507024, −5.52266292023024773035295058494, −5.32367268989330517480770547782, −5.09433930657291656303107411384, −3.90932428186205648874900384635, −3.56071335122131363046138089821, −2.83484721001250786344729583483, −2.07122568382398696308808077558, −1.67010086897332496822657107370, −0.70532577044397818411966240529,
0.70532577044397818411966240529, 1.67010086897332496822657107370, 2.07122568382398696308808077558, 2.83484721001250786344729583483, 3.56071335122131363046138089821, 3.90932428186205648874900384635, 5.09433930657291656303107411384, 5.32367268989330517480770547782, 5.52266292023024773035295058494, 6.41626014943428114089570507024, 6.70950385483033090187267650416, 7.54490658990155895428086736956, 7.86450514138027266808410492180, 7.87806314303922216421800441235, 8.688742893077145323964999997375, 9.396002806365963110901379082504, 9.911162063304479401087819879755, 10.31714912434841002131519526681, 10.54037367682569579920004366535, 11.00305315537488317143596368968
