| L(s) = 1 | + 365·9-s − 1.55e3·11-s + 2.29e3·19-s + 9.84e3·29-s − 3.60e3·31-s − 3.02e4·41-s + 1.34e4·49-s + 6.79e4·59-s + 9.48e4·61-s + 1.50e4·71-s + 1.51e5·79-s + 7.41e4·81-s + 6.11e4·89-s − 5.67e5·99-s − 4.77e4·101-s + 1.41e4·109-s + 1.48e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.88e4·169-s + ⋯ |
| L(s) = 1 | + 1.50·9-s − 3.87·11-s + 1.45·19-s + 2.17·29-s − 0.673·31-s − 2.81·41-s + 0.800·49-s + 2.54·59-s + 3.26·61-s + 0.355·71-s + 2.73·79-s + 1.25·81-s + 0.818·89-s − 5.81·99-s − 0.466·101-s + 0.114·109-s + 9.24·121-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 − 0.104·169-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
| \(L(3)\) |
\(\approx\) |
\(2.736552495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736552495\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - 365 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13450 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 777 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 230 p^{2} T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2838985 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1145 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9435370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4920 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1802 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 34971770 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 15123 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 232488550 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 413370190 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 824735590 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 33960 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47402 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2526616885 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7548 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 567591145 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 75830 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5734483885 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 30585 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6354936190 T^{2} + p^{10} T^{4} \) |
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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
−10.38626227254478507027231720601, −10.37670743847906470872639766521, −9.809672339620648355299120855539, −9.748524593023081797754759346216, −8.612008998714434575185033845183, −8.281945032520183825543402115921, −7.947677496463740998455460505635, −7.48897735530226113829515969757, −6.84904516600385868738954235827, −6.83306167443312266772909931776, −5.58179923653710673871905419780, −5.27294511634304500062273657882, −5.06847744548875266628242237652, −4.49936289110200360425232591089, −3.55964466568951636797552947134, −3.13382610006393151854910150974, −2.33223491205964119807337226066, −2.10608167794618266758825608869, −0.901293353191483212177317457157, −0.50736730807279747250246679695,
0.50736730807279747250246679695, 0.901293353191483212177317457157, 2.10608167794618266758825608869, 2.33223491205964119807337226066, 3.13382610006393151854910150974, 3.55964466568951636797552947134, 4.49936289110200360425232591089, 5.06847744548875266628242237652, 5.27294511634304500062273657882, 5.58179923653710673871905419780, 6.83306167443312266772909931776, 6.84904516600385868738954235827, 7.48897735530226113829515969757, 7.947677496463740998455460505635, 8.281945032520183825543402115921, 8.612008998714434575185033845183, 9.748524593023081797754759346216, 9.809672339620648355299120855539, 10.37670743847906470872639766521, 10.38626227254478507027231720601
