| L(s) = 1 | − 302·9-s − 952·11-s − 7.39e3·23-s + 2.76e3·25-s + 2.78e3·29-s + 2.41e4·37-s − 1.94e4·43-s + 8.62e3·53-s − 4.04e4·67-s − 5.95e4·71-s + 6.63e4·79-s + 3.21e4·81-s + 2.87e5·99-s − 2.51e5·107-s + 1.78e5·109-s − 5.94e4·113-s + 3.57e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.84e5·169-s + ⋯ |
| L(s) = 1 | − 1.24·9-s − 2.37·11-s − 2.91·23-s + 0.885·25-s + 0.615·29-s + 2.90·37-s − 1.60·43-s + 0.421·53-s − 1.10·67-s − 1.40·71-s + 1.19·79-s + 0.544·81-s + 2.94·99-s − 2.12·107-s + 1.43·109-s − 0.437·113-s + 2.22·121-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 + 0.498·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 302 T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2766 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 476 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 184958 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2038210 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4545742 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3696 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1394 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 53548126 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12090 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 77746 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 9724 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 398191362 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4310 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 995172702 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1602855202 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 20236 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 29792 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4018773970 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 33176 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7865547022 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6162094194 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 16858918114 T^{2} + p^{10} T^{4} \) |
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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
−9.322669104387555545816982564285, −8.812218080903069950966984432801, −8.209075378714748043270546395211, −8.206545424331753207156437361232, −7.63079710717712931775191717151, −7.51072185416307254001137709244, −6.37844174457453344865548462060, −6.37389307213136183934631634181, −5.64564787247957064123520540243, −5.51378274470228493187056295521, −4.83512883718446235830102126004, −4.46332764978365459710609150600, −3.81877822341308179765192215791, −3.10890741707897226267911461886, −2.58952927745239023946226473768, −2.48542976310443483781654637334, −1.70826239686442198587932475132, −0.78839241267011651754894935187, 0, 0,
0.78839241267011651754894935187, 1.70826239686442198587932475132, 2.48542976310443483781654637334, 2.58952927745239023946226473768, 3.10890741707897226267911461886, 3.81877822341308179765192215791, 4.46332764978365459710609150600, 4.83512883718446235830102126004, 5.51378274470228493187056295521, 5.64564787247957064123520540243, 6.37389307213136183934631634181, 6.37844174457453344865548462060, 7.51072185416307254001137709244, 7.63079710717712931775191717151, 8.206545424331753207156437361232, 8.209075378714748043270546395211, 8.812218080903069950966984432801, 9.322669104387555545816982564285
