| L(s) = 1 | + 324·5-s + 980·7-s − 810·11-s − 7.88e3·13-s − 2.07e4·17-s − 1.04e4·19-s + 6.05e4·23-s + 2.68e4·25-s + 3.24e4·29-s + 1.45e5·31-s + 3.17e5·35-s − 1.04e4·37-s − 1.08e6·41-s + 6.06e5·43-s + 1.49e6·47-s + 1.28e4·49-s − 2.07e6·53-s − 2.62e5·55-s + 3.07e6·59-s + 6.01e5·61-s − 2.55e6·65-s + 2.42e6·67-s + 3.20e6·71-s + 1.09e6·73-s − 7.93e5·77-s + 1.51e6·79-s + 3.76e6·83-s + ⋯ |
| L(s) = 1 | + 1.15·5-s + 1.07·7-s − 0.183·11-s − 0.994·13-s − 1.02·17-s − 0.349·19-s + 1.03·23-s + 0.344·25-s + 0.246·29-s + 0.880·31-s + 1.25·35-s − 0.0339·37-s − 2.45·41-s + 1.16·43-s + 2.10·47-s + 0.0156·49-s − 1.91·53-s − 0.212·55-s + 1.94·59-s + 0.339·61-s − 1.15·65-s + 0.986·67-s + 1.06·71-s + 0.330·73-s − 0.198·77-s + 0.345·79-s + 0.721·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.369940344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.369940344\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 324 T + 15617 p T^{2} - 324 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 20 p^{2} T + 947505 T^{2} - 20 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 810 T + 37467823 T^{2} + 810 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7880 T + 80881050 T^{2} + 7880 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 20736 T + 761118370 T^{2} + 20736 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10456 T + 90584598 p T^{2} + 10456 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 60588 T + 7560322930 T^{2} - 60588 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 32400 T - 5631978938 T^{2} - 32400 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 145988 T + 43227666033 T^{2} - 145988 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10460 T + 186042173790 T^{2} + 10460 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1085400 T + 666480684862 T^{2} + 1085400 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 606440 T + 304026305298 T^{2} - 606440 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1495908 T + 1499010436642 T^{2} - 1495908 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2079756 T + 3233370268933 T^{2} + 2079756 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3074760 T + 6923204234038 T^{2} - 3074760 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 601120 T + 5712783072042 T^{2} - 601120 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2429240 T + 13016193791970 T^{2} - 2429240 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3201120 T + 18663979591726 T^{2} - 3201120 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1097530 T + 17664142559403 T^{2} - 1097530 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1513160 T + 38619017892318 T^{2} - 1513160 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3760182 T + 55461024499135 T^{2} - 3760182 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10785960 T + 91660099372414 T^{2} - 10785960 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2988910 T + 127894819215315 T^{2} - 2988910 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.12639688312095702453303905003, −9.911074957014213881868145051990, −9.121020271897472293574298637421, −9.070187285709334518340932143812, −8.316481506190661674520527784022, −8.127969490358407671572523926221, −7.25636607537210333602172175109, −7.14230817603457444846035294033, −6.29496510478381467314370199006, −6.20204619704214084998781828327, −5.27521235435411408362088548386, −4.95323449107128764115281600640, −4.77925354838350509569881592247, −3.99008574002095821002547251376, −3.27751850149810749885574448325, −2.51621513455833177480612434129, −2.09542803006166978771204451284, −1.85680711880494829525743008442, −0.917200356104226199079504689366, −0.52590318012716305609494890695,
0.52590318012716305609494890695, 0.917200356104226199079504689366, 1.85680711880494829525743008442, 2.09542803006166978771204451284, 2.51621513455833177480612434129, 3.27751850149810749885574448325, 3.99008574002095821002547251376, 4.77925354838350509569881592247, 4.95323449107128764115281600640, 5.27521235435411408362088548386, 6.20204619704214084998781828327, 6.29496510478381467314370199006, 7.14230817603457444846035294033, 7.25636607537210333602172175109, 8.127969490358407671572523926221, 8.316481506190661674520527784022, 9.070187285709334518340932143812, 9.121020271897472293574298637421, 9.911074957014213881868145051990, 10.12639688312095702453303905003
