| L(s) = 1 | + 232·4-s + 9.22e3·11-s + 1.55e4·16-s + 4.78e4·19-s − 6.93e5·29-s − 2.58e5·31-s + 6.81e5·41-s + 2.13e6·44-s + 3.24e6·49-s + 1.81e6·59-s − 9.56e5·61-s − 1.47e6·64-s + 5.21e6·71-s + 1.11e7·76-s − 1.24e7·79-s + 3.27e7·89-s + 4.55e7·101-s + 3.71e7·109-s − 1.60e8·116-s − 7.22e6·121-s − 5.99e7·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.81·4-s + 2.08·11-s + 0.948·16-s + 1.60·19-s − 5.27·29-s − 1.55·31-s + 1.54·41-s + 3.78·44-s + 3.93·49-s + 1.15·59-s − 0.539·61-s − 0.704·64-s + 1.72·71-s + 2.90·76-s − 2.84·79-s + 4.91·89-s + 4.40·101-s + 2.74·109-s − 9.56·116-s − 0.370·121-s − 2.82·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.525071643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.525071643\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 29 p^{3} T^{2} + 2393 p^{4} T^{4} - 29 p^{17} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3241866 T^{2} + 3983690183803 T^{4} - 3241866 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4612 T + 35518234 T^{2} - 4612 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 188603138 T^{2} + 15884000002783923 T^{4} - 188603138 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 231136540 T^{2} + 109118387602401158 T^{4} - 231136540 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 23934 T + 995928583 T^{2} - 23934 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9685702964 T^{2} + 45905401251825777798 T^{4} - 9685702964 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 346508 T + 64488369118 T^{2} + 346508 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 129178 T + 29398278647 T^{2} + 129178 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 34059096724 T^{2} + \)\(17\!\cdots\!38\)\( T^{4} + 34059096724 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 340928 T + 203171618242 T^{2} - 340928 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 475958888570 T^{2} + \)\(19\!\cdots\!23\)\( T^{4} - 475958888570 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1863978803380 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 1863978803380 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1059553636084 T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - 1059553636084 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 907844 T + 3501429103018 T^{2} - 907844 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 478410 T - 2138250815357 T^{2} + 478410 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14049180666090 T^{2} + \)\(12\!\cdots\!83\)\( T^{4} - 14049180666090 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2607296 T + 17905773132286 T^{2} - 2607296 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 35080618298204 T^{2} + \)\(54\!\cdots\!58\)\( T^{4} - 35080618298204 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6227360 T + 47990625121118 T^{2} + 6227360 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 49505183352548 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 49505183352548 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16356096 T + 133666580484178 T^{2} - 16356096 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 105137761003970 T^{2} + \)\(57\!\cdots\!63\)\( T^{4} - 105137761003970 p^{14} T^{6} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54207835679177143591827688137, −7.33946993485787353406830032825, −7.00725199200654729804502199138, −6.95790183872105342483868290093, −6.85926421634696060856665093354, −5.95720125054541284937205080928, −5.90086933977669158382939758224, −5.87998468446260864477752487576, −5.85974693856740997740802301365, −5.23293624194907815014597858507, −4.95488473810286878307662097836, −4.38036417760117341576378619760, −4.23588650533910130567835760838, −3.71442340291824195646278866801, −3.51610116067441014808590509283, −3.48566962485396724102059713196, −3.18649960416655036298746314304, −2.32212526378494639609993252803, −2.10119271949157578938358413146, −2.07579883552351891157777892608, −1.95937923993564415926605233609, −1.26996731575291439509062214494, −0.968257498990009881973467604268, −0.790692045962651764615952356740, −0.23572036980082100413677619363,
0.23572036980082100413677619363, 0.790692045962651764615952356740, 0.968257498990009881973467604268, 1.26996731575291439509062214494, 1.95937923993564415926605233609, 2.07579883552351891157777892608, 2.10119271949157578938358413146, 2.32212526378494639609993252803, 3.18649960416655036298746314304, 3.48566962485396724102059713196, 3.51610116067441014808590509283, 3.71442340291824195646278866801, 4.23588650533910130567835760838, 4.38036417760117341576378619760, 4.95488473810286878307662097836, 5.23293624194907815014597858507, 5.85974693856740997740802301365, 5.87998468446260864477752487576, 5.90086933977669158382939758224, 5.95720125054541284937205080928, 6.85926421634696060856665093354, 6.95790183872105342483868290093, 7.00725199200654729804502199138, 7.33946993485787353406830032825, 7.54207835679177143591827688137
