| L(s) = 1 | + 162·3-s − 1.40e3·5-s − 3.52e3·7-s + 1.96e4·9-s + 3.12e4·11-s + 2.55e4·13-s − 2.27e5·15-s + 5.13e5·17-s − 4.60e5·19-s − 5.71e5·21-s − 2.89e6·23-s − 1.90e6·25-s + 2.12e6·27-s − 4.07e6·29-s − 1.71e6·31-s + 5.06e6·33-s + 4.95e6·35-s − 1.49e7·37-s + 4.13e6·39-s − 3.01e6·41-s + 2.26e7·43-s − 2.76e7·45-s − 4.12e6·47-s − 4.55e7·49-s + 8.31e7·51-s − 1.42e8·53-s − 4.38e7·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.00·5-s − 0.555·7-s + 9-s + 0.643·11-s + 0.247·13-s − 1.16·15-s + 1.49·17-s − 0.810·19-s − 0.641·21-s − 2.15·23-s − 0.973·25-s + 0.769·27-s − 1.06·29-s − 0.333·31-s + 0.742·33-s + 0.557·35-s − 1.31·37-s + 0.286·39-s − 0.166·41-s + 1.01·43-s − 1.00·45-s − 0.123·47-s − 1.12·49-s + 1.72·51-s − 2.47·53-s − 0.646·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 1404 T + 154878 p^{2} T^{2} + 1404 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 72 p^{2} T + 1183486 p^{2} T^{2} + 72 p^{11} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2840 p T + 690337382 T^{2} - 2840 p^{10} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1964 p T + 707384286 T^{2} - 1964 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 513396 T + 207412890694 T^{2} - 513396 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 460296 T + 284568965878 T^{2} + 460296 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2895744 T + 5698467869614 T^{2} + 2895744 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4070268 T + 22104145749630 T^{2} + 4070268 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1714104 T + 34250236970062 T^{2} + 1714104 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14934340 T + 274677422189454 T^{2} + 14934340 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3016188 T + 386430948236502 T^{2} + 3016188 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 22696200 T + 865645551135910 T^{2} - 22696200 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4128880 T + 1610984272411550 T^{2} + 4128880 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 142441452 T + 11625512396236686 T^{2} + 142441452 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 53974408 T + 18027519514607558 T^{2} + 53974408 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 255401652 T + 39646728248616542 T^{2} + 255401652 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12869496 T + 38882381301235798 T^{2} + 12869496 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 42832288 T + 91424577036976142 T^{2} + 42832288 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 637706124 T + 214912498624320566 T^{2} + 637706124 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 744065208 T + 373311879141585838 T^{2} + 744065208 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 302340120 T + 225775924734463990 T^{2} - 302340120 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 976469868 T + 644973131913633174 T^{2} + 976469868 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2445646108 T + 3008618100133008774 T^{2} + 2445646108 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.98466948269273820665223789674, −11.48021770716855801001113508942, −10.74062486463409934374778405443, −10.11153221439132482568205931949, −9.592623798740150076533387344020, −9.241308740927539838052578901953, −8.363831940298558001002421769383, −8.068445412948300156647676295363, −7.56355941985985021257603760518, −7.05331090039734695083592511139, −6.05242809261410513344965872733, −5.78900753338508179067707376005, −4.26641735622107401858808752882, −4.23499575491122279062371133198, −3.26346341270778658774864748455, −3.18071138264807310742495310199, −1.71478756100922604862871972476, −1.64446256196138061347470164116, 0, 0,
1.64446256196138061347470164116, 1.71478756100922604862871972476, 3.18071138264807310742495310199, 3.26346341270778658774864748455, 4.23499575491122279062371133198, 4.26641735622107401858808752882, 5.78900753338508179067707376005, 6.05242809261410513344965872733, 7.05331090039734695083592511139, 7.56355941985985021257603760518, 8.068445412948300156647676295363, 8.363831940298558001002421769383, 9.241308740927539838052578901953, 9.592623798740150076533387344020, 10.11153221439132482568205931949, 10.74062486463409934374778405443, 11.48021770716855801001113508942, 11.98466948269273820665223789674
