| L(s) = 1 | + 14·3-s − 294·5-s − 686·7-s − 3.21e3·9-s − 3.49e3·11-s − 1.61e4·13-s − 4.11e3·15-s − 2.92e4·17-s − 3.20e3·19-s − 9.60e3·21-s − 9.36e3·23-s + 3.06e4·25-s − 6.22e4·27-s + 1.84e5·29-s + 1.65e5·31-s − 4.88e4·33-s + 2.01e5·35-s + 2.86e5·37-s − 2.26e5·39-s − 1.16e5·41-s − 2.94e5·43-s + 9.46e5·45-s − 1.01e6·47-s + 3.52e5·49-s − 4.09e5·51-s − 1.39e6·53-s + 1.02e6·55-s + ⋯ |
| L(s) = 1 | + 0.299·3-s − 1.05·5-s − 0.755·7-s − 1.47·9-s − 0.791·11-s − 2.04·13-s − 0.314·15-s − 1.44·17-s − 0.107·19-s − 0.226·21-s − 0.160·23-s + 0.392·25-s − 0.608·27-s + 1.40·29-s + 0.995·31-s − 0.236·33-s + 0.795·35-s + 0.928·37-s − 0.611·39-s − 0.264·41-s − 0.564·43-s + 1.54·45-s − 1.42·47-s + 3/7·49-s − 0.432·51-s − 1.28·53-s + 0.832·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 1138 p T^{2} - 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 294 T + 11154 p T^{2} + 294 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3492 T + 8600742 T^{2} + 3492 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 16170 T + 150099650 T^{2} + 16170 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 29232 T + 887067486 T^{2} + 29232 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3206 T + 330079206 T^{2} + 3206 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9360 T + 5917092558 T^{2} + 9360 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 184704 T + 39838908966 T^{2} - 184704 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 165060 T + 48364547678 T^{2} - 165060 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 286144 T + 144034750326 T^{2} - 286144 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 116760 T + 300275715646 T^{2} + 116760 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 294428 T + 533824957446 T^{2} + 294428 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1014132 T + 1229625493438 T^{2} + 1014132 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1396452 T + 2574488431294 T^{2} + 1396452 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2729286 T + 6810337729638 T^{2} + 2729286 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2466954 T + 6172226893562 T^{2} - 2466954 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 225176 T + 3298823282406 T^{2} - 225176 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1530312 T + 12745388415342 T^{2} - 1530312 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1143548 T + 2698400643174 T^{2} - 1143548 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7951176 T + 50346024049886 T^{2} - 7951176 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18487854 T + 138806969812134 T^{2} + 18487854 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4652508 T + 76401824909398 T^{2} + 4652508 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26702368 T + 336379902255582 T^{2} + 26702368 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−15.11105203095668464002805776665, −15.05569900760741093739859203013, −14.06701542246417433814737309472, −13.62361943165911306943334020314, −12.62511458099424490990923546578, −12.29229899146141975387039684831, −11.47065165084101464662263004385, −11.02766854584370790829776601718, −9.968615970845589997386328613737, −9.478803074990492011641460549072, −8.282814878331397264335691782473, −8.193348197772232574703031973277, −7.08165516485800333005193773450, −6.36462744902072532906910942325, −5.13008884510804739598291877116, −4.38166338862504466256642613888, −2.95722235374892833709467497848, −2.57095458641617698954174958071, 0, 0,
2.57095458641617698954174958071, 2.95722235374892833709467497848, 4.38166338862504466256642613888, 5.13008884510804739598291877116, 6.36462744902072532906910942325, 7.08165516485800333005193773450, 8.193348197772232574703031973277, 8.282814878331397264335691782473, 9.478803074990492011641460549072, 9.968615970845589997386328613737, 11.02766854584370790829776601718, 11.47065165084101464662263004385, 12.29229899146141975387039684831, 12.62511458099424490990923546578, 13.62361943165911306943334020314, 14.06701542246417433814737309472, 15.05569900760741093739859203013, 15.11105203095668464002805776665
