| L(s) = 1 | + 482·9-s − 3.82e3·17-s − 6.25e3·25-s + 2.78e4·41-s − 3.36e4·49-s + 1.00e5·73-s + 1.73e5·81-s + 1.44e4·89-s − 1.70e5·97-s − 5.12e5·113-s + 9.74e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.84e6·153-s + 157-s + 163-s + 167-s − 7.42e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.98·9-s − 3.21·17-s − 2·25-s + 2.58·41-s − 2·49-s + 2.21·73-s + 2.93·81-s + 0.193·89-s − 1.84·97-s − 3.77·113-s + 0.604·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 6.37·153-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 2·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.649591581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649591581\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 482 T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 97426 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 1914 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 3353726 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 13926 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 214485614 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 921043598 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1813708382 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 50402 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 96051518 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7218 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 85450 T + p^{5} T^{2} )^{2} \) |
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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
−11.09886135470749457968716234533, −11.00944991136892335059074176011, −10.62142138051592028826575456843, −9.644845183245651639668173970727, −9.608165764034525858391410918055, −9.254466977321885676733254455685, −8.393504868117517563491176566064, −7.985058410219326367726379831171, −7.39730964946385004169750971344, −6.89079854682615770548709827564, −6.49703438551855314091937712399, −6.02887120166681121519450489449, −5.10612179965851915162786342932, −4.40244569150887977146047966245, −4.26107039643747058517446495806, −3.67894920363781064059128065000, −2.43654980201822610444196989105, −2.06157902314429723389353765820, −1.36956016796031719597627736737, −0.36131237912144818287646933590,
0.36131237912144818287646933590, 1.36956016796031719597627736737, 2.06157902314429723389353765820, 2.43654980201822610444196989105, 3.67894920363781064059128065000, 4.26107039643747058517446495806, 4.40244569150887977146047966245, 5.10612179965851915162786342932, 6.02887120166681121519450489449, 6.49703438551855314091937712399, 6.89079854682615770548709827564, 7.39730964946385004169750971344, 7.985058410219326367726379831171, 8.393504868117517563491176566064, 9.254466977321885676733254455685, 9.608165764034525858391410918055, 9.644845183245651639668173970727, 10.62142138051592028826575456843, 11.00944991136892335059074176011, 11.09886135470749457968716234533
