| L(s) = 1 | + 294·5-s − 686·7-s + 3.49e3·11-s − 1.61e4·13-s + 2.92e4·17-s − 3.20e3·19-s + 9.36e3·23-s + 3.06e4·25-s − 1.84e5·29-s + 1.65e5·31-s − 2.01e5·35-s + 2.86e5·37-s + 1.16e5·41-s − 2.94e5·43-s + 1.01e6·47-s + 3.52e5·49-s + 1.39e6·53-s + 1.02e6·55-s + 2.72e6·59-s + 2.46e6·61-s − 4.75e6·65-s + 2.25e5·67-s − 1.53e6·71-s + 1.14e6·73-s − 2.39e6·77-s + 7.95e6·79-s + 1.84e7·83-s + ⋯ |
| L(s) = 1 | + 1.05·5-s − 0.755·7-s + 0.791·11-s − 2.04·13-s + 1.44·17-s − 0.107·19-s + 0.160·23-s + 0.392·25-s − 1.40·29-s + 0.995·31-s − 0.795·35-s + 0.928·37-s + 0.264·41-s − 0.564·43-s + 1.42·47-s + 3/7·49-s + 1.28·53-s + 0.832·55-s + 1.73·59-s + 1.39·61-s − 2.14·65-s + 0.0914·67-s − 0.507·71-s + 0.344·73-s − 0.597·77-s + 1.81·79-s + 3.54·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.387725424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.387725424\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 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
−10.95198485763641273762582184256, −10.27535757222391192689818082499, −9.988605190591271416453074024766, −9.712273205471157723709648294335, −9.210150752017228469174881210227, −8.924815097104136382331295828057, −7.84999593016287602422788543907, −7.74154876063181111895127284993, −6.80646036037028168159436303779, −6.78317236733984638802734821033, −5.91787430576143054860462718506, −5.48496527292089878826835550753, −5.11649637882978872494397265467, −4.27888304572557853056040212740, −3.67982727526506621342290639050, −3.01803973324688055718927750540, −2.24207696487751290049713024127, −2.07049225056689533275357687962, −0.858754003746904079424637509985, −0.62748393223175989382611561939,
0.62748393223175989382611561939, 0.858754003746904079424637509985, 2.07049225056689533275357687962, 2.24207696487751290049713024127, 3.01803973324688055718927750540, 3.67982727526506621342290639050, 4.27888304572557853056040212740, 5.11649637882978872494397265467, 5.48496527292089878826835550753, 5.91787430576143054860462718506, 6.78317236733984638802734821033, 6.80646036037028168159436303779, 7.74154876063181111895127284993, 7.84999593016287602422788543907, 8.924815097104136382331295828057, 9.210150752017228469174881210227, 9.712273205471157723709648294335, 9.988605190591271416453074024766, 10.27535757222391192689818082499, 10.95198485763641273762582184256
