| L(s) = 1 | − 36·5-s − 120·7-s + 200·11-s + 284·13-s − 2.67e3·17-s + 72·19-s − 3.84e3·23-s + 2.65e3·25-s − 1.02e4·29-s + 1.04e4·31-s + 4.32e3·35-s + 1.31e4·37-s − 4.16e3·41-s − 5.83e3·43-s − 1.52e3·47-s − 1.48e4·49-s − 9.01e3·53-s − 7.20e3·55-s + 5.50e4·59-s − 6.34e4·61-s − 1.02e4·65-s + 3.67e4·67-s − 3.76e4·71-s − 3.78e4·73-s − 2.40e4·77-s + 1.44e5·79-s + 1.09e5·83-s + ⋯ |
| L(s) = 1 | − 0.643·5-s − 0.925·7-s + 0.498·11-s + 0.466·13-s − 2.24·17-s + 0.0457·19-s − 1.51·23-s + 0.850·25-s − 2.25·29-s + 1.96·31-s + 0.596·35-s + 1.57·37-s − 0.386·41-s − 0.481·43-s − 0.100·47-s − 0.885·49-s − 0.440·53-s − 0.320·55-s + 2.06·59-s − 2.18·61-s − 0.300·65-s + 1.00·67-s − 0.886·71-s − 0.830·73-s − 0.461·77-s + 2.61·79-s + 1.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2590648598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2590648598\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 36 T - 1362 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 120 T + 29278 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 200 T + 46406 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 284 T + 477054 T^{2} - 284 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2676 T + 4344262 T^{2} + 2676 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4159894 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3840 T + 9416686 T^{2} + 3840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10212 T + 64228638 T^{2} + 10212 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10488 T + 84559438 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13148 T + 125908974 T^{2} - 13148 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4164 T + 216207126 T^{2} + 4164 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5832 T + 168401542 T^{2} + 5832 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1520 T + 148144670 T^{2} + 1520 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9012 T + 816689646 T^{2} + 9012 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 55096 T + 1818478886 T^{2} - 55096 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 63444 T + 2677193342 T^{2} + 63444 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 36792 T + 1148752246 T^{2} - 36792 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 37664 T + 2440628942 T^{2} + 37664 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37836 T + 2085902966 T^{2} + 37836 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 144888 T + 10711615534 T^{2} - 144888 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 109272 T + 10472055958 T^{2} - 109272 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 32556 T + 8836559958 T^{2} - 32556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 69092 T + 18339537030 T^{2} - 69092 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.10349898750354388844759098873, −10.99592506406160479889289097081, −10.15761506248492035931915018225, −9.813597982188160246565459516624, −9.207721128276852356687927329682, −8.951303769847119867169886672344, −8.304242936937685415001827320874, −7.88489096029691078771304770369, −7.32347867296518986850370489202, −6.62992318519195227892801964667, −6.25151844223162851296684755208, −6.10173524444721992750269603546, −4.91926282841069563636886183455, −4.57694241511532075752866878131, −3.71737995047683625391745563519, −3.67160078121328846309029964919, −2.59405891947027587264839096536, −2.09299084263045959209608581651, −1.11808101953705506783798796545, −0.15018176816877938014164731810,
0.15018176816877938014164731810, 1.11808101953705506783798796545, 2.09299084263045959209608581651, 2.59405891947027587264839096536, 3.67160078121328846309029964919, 3.71737995047683625391745563519, 4.57694241511532075752866878131, 4.91926282841069563636886183455, 6.10173524444721992750269603546, 6.25151844223162851296684755208, 6.62992318519195227892801964667, 7.32347867296518986850370489202, 7.88489096029691078771304770369, 8.304242936937685415001827320874, 8.951303769847119867169886672344, 9.207721128276852356687927329682, 9.813597982188160246565459516624, 10.15761506248492035931915018225, 10.99592506406160479889289097081, 11.10349898750354388844759098873
