| L(s) = 1 | + 14·5-s + 2·13-s − 48·17-s + 97·25-s − 110·29-s − 96·37-s − 14·41-s + 73·49-s − 96·53-s + 50·61-s + 28·65-s − 240·73-s − 672·85-s + 240·89-s + 50·97-s − 226·101-s + 384·109-s + 62·113-s + 73·121-s + 322·125-s + 127-s + 131-s + 137-s + 139-s − 1.54e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 14/5·5-s + 2/13·13-s − 2.82·17-s + 3.87·25-s − 3.79·29-s − 2.59·37-s − 0.341·41-s + 1.48·49-s − 1.81·53-s + 0.819·61-s + 0.430·65-s − 3.28·73-s − 7.90·85-s + 2.69·89-s + 0.515·97-s − 2.23·101-s + 3.52·109-s + 0.548·113-s + 0.603·121-s + 2.57·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 10.6·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.439332967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439332967\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 889 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1561 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 889 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3049 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1223 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10078 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 743 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10297 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
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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
−9.016258648777085013249053190667, −8.705673813261645087981496266877, −8.471633933726061115793225361219, −7.46219425142435923383794327317, −7.35040191348832522744662684515, −6.92682225575655658307079098111, −6.41882155227491800715247234735, −6.22417878653494108094028807483, −5.75510985103102111587691446093, −5.52798830441229429611393654847, −5.15902610815244125552624939080, −4.64534837826805113660716357207, −4.21627316310507311403045791128, −3.56410805130167536089994989162, −3.16611182062227909117784182743, −2.20958068998342900185009374753, −2.20102345366081046198274559196, −1.81604618047391818424846680074, −1.46516861329928072143458554390, −0.24137755821062120576700217033,
0.24137755821062120576700217033, 1.46516861329928072143458554390, 1.81604618047391818424846680074, 2.20102345366081046198274559196, 2.20958068998342900185009374753, 3.16611182062227909117784182743, 3.56410805130167536089994989162, 4.21627316310507311403045791128, 4.64534837826805113660716357207, 5.15902610815244125552624939080, 5.52798830441229429611393654847, 5.75510985103102111587691446093, 6.22417878653494108094028807483, 6.41882155227491800715247234735, 6.92682225575655658307079098111, 7.35040191348832522744662684515, 7.46219425142435923383794327317, 8.471633933726061115793225361219, 8.705673813261645087981496266877, 9.016258648777085013249053190667
