| L(s) = 1 | + 14·9-s + 32·13-s + 20·17-s + 48·29-s − 136·37-s − 188·41-s + 76·49-s + 88·53-s + 152·61-s − 356·73-s − 15·81-s − 276·89-s + 360·97-s + 296·101-s + 40·109-s + 220·113-s + 448·117-s + 334·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 280·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 14/9·9-s + 2.46·13-s + 1.17·17-s + 1.65·29-s − 3.67·37-s − 4.58·41-s + 1.55·49-s + 1.66·53-s + 2.49·61-s − 4.87·73-s − 0.185·81-s − 3.10·89-s + 3.71·97-s + 2.93·101-s + 0.366·109-s + 1.94·113-s + 3.82·117-s + 2.76·121-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 + 1.83·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.593937928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.593937928\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2}( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 10 p^{3} T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 334 T^{2} + 54355 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 10 T - 101 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 623 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1900 T^{2} + 1450918 T^{4} - 1900 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 242 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2540 T^{2} + 3054438 T^{4} - 2540 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 68 T + 3718 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 94 T + 4867 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 772 T^{2} + 4778854 T^{4} - 772 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 1358 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 44 T + 5398 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12004 T^{2} + 59537830 T^{4} - 12004 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 76 T + 4486 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6206 T^{2} + 30061155 T^{4} - 6206 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 5924 T^{2} + 39727110 T^{4} - 5924 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 89 T + p^{2} T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2572 T^{2} + 28232358 T^{4} - 2572 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 19406 T^{2} + 173224851 T^{4} - 19406 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 138 T + 14267 T^{2} + 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 180 T + 20582 T^{2} - 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.73237101108553626793465822440, −6.25668616078356403226158262368, −6.09111617526296710286903146914, −5.77320675711103907744878681644, −5.62192511511135750316598901571, −5.60363710323744447121517506589, −5.19723429710127477944933467440, −4.96727552034768516327247156807, −4.65466924770437307725466885389, −4.53715195975446117138337821243, −4.45756793086298428209672628982, −3.81835383693732636928130566048, −3.69947733558054654478973658850, −3.59648811824811599006787742114, −3.50251121915346200562622064407, −3.23077669335504322356242015667, −2.84821743438083102352470543718, −2.58179699514104487761660575748, −2.08752522220716111457262143141, −1.67596526023742566808064319644, −1.63010691060332921135293866961, −1.46014598224741156716846189563, −1.05611810882230524172528911174, −0.71421372394960255822118943882, −0.31900081191940336776390074543,
0.31900081191940336776390074543, 0.71421372394960255822118943882, 1.05611810882230524172528911174, 1.46014598224741156716846189563, 1.63010691060332921135293866961, 1.67596526023742566808064319644, 2.08752522220716111457262143141, 2.58179699514104487761660575748, 2.84821743438083102352470543718, 3.23077669335504322356242015667, 3.50251121915346200562622064407, 3.59648811824811599006787742114, 3.69947733558054654478973658850, 3.81835383693732636928130566048, 4.45756793086298428209672628982, 4.53715195975446117138337821243, 4.65466924770437307725466885389, 4.96727552034768516327247156807, 5.19723429710127477944933467440, 5.60363710323744447121517506589, 5.62192511511135750316598901571, 5.77320675711103907744878681644, 6.09111617526296710286903146914, 6.25668616078356403226158262368, 6.73237101108553626793465822440
