| L(s) = 1 | − 9·9-s + 96·11-s + 280·19-s − 420·29-s − 544·31-s − 396·41-s + 670·49-s + 480·59-s + 604·61-s + 1.53e3·71-s − 1.28e3·79-s + 81·81-s − 420·89-s − 864·99-s + 3.44e3·101-s + 1.22e3·109-s + 4.25e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.63·11-s + 3.38·19-s − 2.68·29-s − 3.15·31-s − 1.50·41-s + 1.95·49-s + 1.05·59-s + 1.26·61-s + 2.56·71-s − 1.82·79-s + 1/9·81-s − 0.500·89-s − 0.877·99-s + 3.39·101-s + 1.07·109-s + 3.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.258995545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.258995545\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4390 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 272 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 10250 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 87190 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 160990 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291670 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 302 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 246310 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 549550 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 640 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1022470 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 527810 T^{2} + p^{6} T^{4} \) |
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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
−9.359799366833261840636254597388, −9.312330133089882012630771800149, −8.871559736343740868445159615325, −8.605876664799966058516507915818, −7.72943548797226142481689761762, −7.44456820201042208779106791273, −7.05710802281960569950249482998, −6.94910197208657226420546753877, −6.19560886629945639039652644543, −5.63641496991596172262758155199, −5.34628656762851323706343060284, −5.23391451551795439624459672109, −4.05721486601111057134018429552, −3.92338759499866711659654893239, −3.32786197715328077176298711000, −3.31210587949333384787610363608, −1.94103194222554993707145549004, −1.83957999922518805068485831117, −1.05292629844596158041362267066, −0.56398853839584598481768331207,
0.56398853839584598481768331207, 1.05292629844596158041362267066, 1.83957999922518805068485831117, 1.94103194222554993707145549004, 3.31210587949333384787610363608, 3.32786197715328077176298711000, 3.92338759499866711659654893239, 4.05721486601111057134018429552, 5.23391451551795439624459672109, 5.34628656762851323706343060284, 5.63641496991596172262758155199, 6.19560886629945639039652644543, 6.94910197208657226420546753877, 7.05710802281960569950249482998, 7.44456820201042208779106791273, 7.72943548797226142481689761762, 8.605876664799966058516507915818, 8.871559736343740868445159615325, 9.312330133089882012630771800149, 9.359799366833261840636254597388
