| L(s) = 1 | + 1.00e3·4-s + 1.72e5·11-s + 7.53e5·16-s − 1.43e6·19-s − 6.38e6·29-s − 4.69e6·31-s + 5.85e7·41-s + 1.74e8·44-s + 2.17e7·49-s + 1.21e8·59-s − 2.53e8·61-s + 4.95e8·64-s + 3.51e8·71-s − 1.44e9·76-s − 4.68e8·79-s − 6.33e8·89-s + 1.30e9·101-s + 1.86e9·109-s − 6.43e9·116-s + 1.76e10·121-s − 4.73e9·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.96·4-s + 3.55·11-s + 2.87·16-s − 2.52·19-s − 1.67·29-s − 0.913·31-s + 3.23·41-s + 7.00·44-s + 0.538·49-s + 1.30·59-s − 2.34·61-s + 3.69·64-s + 1.63·71-s − 4.96·76-s − 1.35·79-s − 1.06·89-s + 1.25·101-s + 1.26·109-s − 3.30·116-s + 7.49·121-s − 1.79·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.270086383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.270086383\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 63 p^{4} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 21724814 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 86404 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1284401738 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 194068858110 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 716284 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1725624516526 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3194402 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2349000 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 91103349613946 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 29282630 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1002884772181510 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2237881795680030 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6576989091999366 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 60616076 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 126745682 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 42051486686836790 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 175551608 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 113993650264313326 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 234431160 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 359740830160770262 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 316534326 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1461455752222471870 T^{2} + p^{18} T^{4} \) |
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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
−11.02446512781336997865724112360, −10.65981164474762885754600570163, −9.853406882625657095197250942862, −9.413418267996648451049760721358, −8.852772693681937122748731535332, −8.600707122103742163632789795270, −7.54304514544516979353807668168, −7.37942136923554637505514196724, −6.74352036254651512650312980855, −6.29691999929754003733322518494, −6.15445857889417270276188540624, −5.63248541578336711116012634397, −4.32044408965345829776686262395, −3.99721471299958580141394183352, −3.64456909902492691892661165326, −2.78249443332627109442736792174, −1.92838684942071060986755102368, −1.89963845105653498786930770886, −1.21114987802860942644848370986, −0.61148901169788849291609049794,
0.61148901169788849291609049794, 1.21114987802860942644848370986, 1.89963845105653498786930770886, 1.92838684942071060986755102368, 2.78249443332627109442736792174, 3.64456909902492691892661165326, 3.99721471299958580141394183352, 4.32044408965345829776686262395, 5.63248541578336711116012634397, 6.15445857889417270276188540624, 6.29691999929754003733322518494, 6.74352036254651512650312980855, 7.37942136923554637505514196724, 7.54304514544516979353807668168, 8.600707122103742163632789795270, 8.852772693681937122748731535332, 9.413418267996648451049760721358, 9.853406882625657095197250942862, 10.65981164474762885754600570163, 11.02446512781336997865724112360
