| L(s) = 1 | + 342·9-s − 1.08e3·11-s + 1.67e3·19-s + 1.18e3·29-s − 8.51e3·31-s + 3.44e4·41-s + 2.58e4·49-s − 1.53e4·59-s − 6.94e4·61-s + 9.37e4·71-s − 1.53e5·79-s + 5.79e4·81-s − 5.95e4·89-s − 3.69e5·99-s + 2.25e4·101-s − 1.99e5·109-s + 5.52e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.67e5·169-s + ⋯ |
| L(s) = 1 | + 1.40·9-s − 2.69·11-s + 1.06·19-s + 0.262·29-s − 1.59·31-s + 3.20·41-s + 1.53·49-s − 0.573·59-s − 2.39·61-s + 2.20·71-s − 2.77·79-s + 0.980·81-s − 0.796·89-s − 3.78·99-s + 0.220·101-s − 1.61·109-s + 3.43·121-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 + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.52·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.796719137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796719137\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 38 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 25870 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 540 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 567862 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2486878 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 44 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3970130 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 594 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4256 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138599110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 17226 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 147606886 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 457010398 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 456374950 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7668 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 34738 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2224486870 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 418480658 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 76912 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3292624630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 29754 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2193410110 T^{2} + p^{10} T^{4} \) |
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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
−10.62475785789231264987025512885, −10.24305275977156155804027829045, −9.943082018858907392505444495749, −9.190305899352389207720420107799, −9.149797018212453839295726719301, −8.193925878898288663955390245380, −7.73481572594849816144466861808, −7.39088717216677277379639610084, −7.35170991948145031567044906432, −6.43654625103470047892974304877, −5.71146336289234190997462984024, −5.44232128880657713125432632030, −4.92557223846988447624193660257, −4.29457476514301219585111220462, −3.85275774078253835014305425309, −2.81058718284539763051121904714, −2.70282080497535337503635024705, −1.82828414998634702990851181103, −1.10498373110026333324638960889, −0.35694795194284638212864983865,
0.35694795194284638212864983865, 1.10498373110026333324638960889, 1.82828414998634702990851181103, 2.70282080497535337503635024705, 2.81058718284539763051121904714, 3.85275774078253835014305425309, 4.29457476514301219585111220462, 4.92557223846988447624193660257, 5.44232128880657713125432632030, 5.71146336289234190997462984024, 6.43654625103470047892974304877, 7.35170991948145031567044906432, 7.39088717216677277379639610084, 7.73481572594849816144466861808, 8.193925878898288663955390245380, 9.149797018212453839295726719301, 9.190305899352389207720420107799, 9.943082018858907392505444495749, 10.24305275977156155804027829045, 10.62475785789231264987025512885
