| L(s) = 1 | + 8·9-s + 16·11-s + 20·19-s − 4·49-s + 400·61-s − 114·81-s + 128·99-s + 16·101-s − 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 424·169-s + 160·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/9·9-s + 1.45·11-s + 1.05·19-s − 0.0816·49-s + 6.55·61-s − 1.40·81-s + 1.29·99-s + 0.158·101-s − 2.67·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.50·169-s + 0.935·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06206392479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06206392479\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 212 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 338 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2}( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1604 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2018 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1006 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4322 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2468 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 1586 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 100 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8852 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8738 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 10274 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 386 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 12914 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 5662 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 3572 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.50441956100096602615952649546, −6.43073395322092381783298602363, −5.69760065346460010172508411576, −5.66286948546502727403214849648, −5.61479893756429847183601068491, −5.38261454771521391094404404423, −4.99848446232432113818303485637, −4.99651069815972846043679776409, −4.58392839608896344767036712364, −4.33048856393496567591629527446, −4.15225107781526813745038783942, −3.85147787726749751525558779307, −3.82116638962625667236239530692, −3.60743680221930865271393667519, −3.26109704006816808844018487537, −3.06190390045238083023835253805, −2.64522076764186682551414233619, −2.29864700018671812793133930127, −2.28612035555357132984621514551, −1.91141159342762503808870248590, −1.49646195132628934564199736437, −1.07255887754472467041908348877, −0.987822757220466272305843052073, −0.957319116968392446389957088097, −0.02566179177811209413519894636,
0.02566179177811209413519894636, 0.957319116968392446389957088097, 0.987822757220466272305843052073, 1.07255887754472467041908348877, 1.49646195132628934564199736437, 1.91141159342762503808870248590, 2.28612035555357132984621514551, 2.29864700018671812793133930127, 2.64522076764186682551414233619, 3.06190390045238083023835253805, 3.26109704006816808844018487537, 3.60743680221930865271393667519, 3.82116638962625667236239530692, 3.85147787726749751525558779307, 4.15225107781526813745038783942, 4.33048856393496567591629527446, 4.58392839608896344767036712364, 4.99651069815972846043679776409, 4.99848446232432113818303485637, 5.38261454771521391094404404423, 5.61479893756429847183601068491, 5.66286948546502727403214849648, 5.69760065346460010172508411576, 6.43073395322092381783298602363, 6.50441956100096602615952649546
