| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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
−9.639337217953820013893886595872, −9.474603821789770704200280076551, −8.696460955436759256457652153788, −8.244811677651075042702015886873, −7.79063894621319039104693593755, −7.72643297703543015251777522277, −7.17706431692752032389758911744, −6.55841047118552127080844812474, −5.89889859553937841763308379621, −5.51413105946767965968711406769, −4.76775883544457608787092559280, −4.51108098762543528532472144388, −3.93176185443377936062755255446, −3.55861448718882447172055426018, −2.49447153085580313612443392678, −2.48815880716351009272307847460, −1.31185342272342231078399438909, −1.14468091903624790257576966398, 0, 0,
1.14468091903624790257576966398, 1.31185342272342231078399438909, 2.48815880716351009272307847460, 2.49447153085580313612443392678, 3.55861448718882447172055426018, 3.93176185443377936062755255446, 4.51108098762543528532472144388, 4.76775883544457608787092559280, 5.51413105946767965968711406769, 5.89889859553937841763308379621, 6.55841047118552127080844812474, 7.17706431692752032389758911744, 7.72643297703543015251777522277, 7.79063894621319039104693593755, 8.244811677651075042702015886873, 8.696460955436759256457652153788, 9.474603821789770704200280076551, 9.639337217953820013893886595872
