| L(s) = 1 | − 16·2-s + 45·3-s + 192·4-s + 250·5-s − 720·6-s − 686·7-s − 2.04e3·8-s − 2.50e3·9-s − 4.00e3·10-s − 2.84e3·11-s + 8.64e3·12-s + 753·13-s + 1.09e4·14-s + 1.12e4·15-s + 2.04e4·16-s − 6.41e3·17-s + 4.00e4·18-s + 2.50e3·19-s + 4.80e4·20-s − 3.08e4·21-s + 4.55e4·22-s − 3.11e4·23-s − 9.21e4·24-s + 4.68e4·25-s − 1.20e4·26-s − 2.18e5·27-s − 1.31e5·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.962·3-s + 3/2·4-s + 0.894·5-s − 1.36·6-s − 0.755·7-s − 1.41·8-s − 1.14·9-s − 1.26·10-s − 0.644·11-s + 1.44·12-s + 0.0950·13-s + 1.06·14-s + 0.860·15-s + 5/4·16-s − 0.316·17-s + 1.61·18-s + 0.0836·19-s + 1.34·20-s − 0.727·21-s + 0.911·22-s − 0.533·23-s − 1.36·24-s + 3/5·25-s − 0.134·26-s − 2.13·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5 p^{2} T + 1510 p T^{2} - 5 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2845 T - 6692542 T^{2} + 2845 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 753 T + 77948396 T^{2} - 753 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6413 T + 58578932 T^{2} + 6413 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2502 T + 1748341838 T^{2} - 2502 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 31158 T + 461266174 T^{2} + 31158 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 72699 T + 27821734708 T^{2} + 72699 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10472 p T + 64287828414 T^{2} + 10472 p^{8} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 776108 T + 340404268782 T^{2} + 776108 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 935222 T + 600319227794 T^{2} + 935222 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 727354 T + 506716055022 T^{2} + 727354 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 916211 T + 1020403468094 T^{2} + 916211 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 29606 T - 543261749398 T^{2} - 29606 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1596008 T + 5092471155638 T^{2} - 1596008 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 109862 T + 6102866214282 T^{2} - 109862 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4832652 T + 17896221486422 T^{2} - 4832652 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1051952 T + 18266757793454 T^{2} + 1051952 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3819128 T + 23764134405534 T^{2} + 3819128 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6266441 T + 25936515528558 T^{2} - 6266441 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15296600 T + 107991709313990 T^{2} + 15296600 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1772750 T + 12446633718458 T^{2} - 1772750 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5483489 T + 115007232513276 T^{2} + 5483489 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−13.10609758522797859771256821079, −12.42206162447943073109168328261, −11.58563045935552238804058549910, −11.22657382225733352169505231375, −10.29298500099118194441680217492, −10.14592780685252230672728943790, −9.327596019227754802425382397941, −9.024684510819137764085553368750, −8.324708806454048781551817782678, −8.180956523572888834329760375498, −6.98673042660931856733980110776, −6.75315662679899539543134748377, −5.57794565451507156139192684169, −5.43569235955916930903886967872, −3.41538613585248844474751170471, −3.19234029650960505055856048384, −2.08826371578148989409550834858, −1.80203597040632905700581340077, 0, 0,
1.80203597040632905700581340077, 2.08826371578148989409550834858, 3.19234029650960505055856048384, 3.41538613585248844474751170471, 5.43569235955916930903886967872, 5.57794565451507156139192684169, 6.75315662679899539543134748377, 6.98673042660931856733980110776, 8.180956523572888834329760375498, 8.324708806454048781551817782678, 9.024684510819137764085553368750, 9.327596019227754802425382397941, 10.14592780685252230672728943790, 10.29298500099118194441680217492, 11.22657382225733352169505231375, 11.58563045935552238804058549910, 12.42206162447943073109168328261, 13.10609758522797859771256821079
