| L(s) = 1 | + 9·2-s + 71·4-s + 360·5-s + 686·7-s + 1.70e3·8-s + 3.24e3·10-s + 4.93e3·11-s + 7.70e3·13-s + 6.17e3·14-s + 8.58e3·16-s + 2.85e4·17-s − 6.37e4·19-s + 2.55e4·20-s + 4.43e4·22-s − 8.22e4·23-s + 4.74e4·25-s + 6.93e4·26-s + 4.87e4·28-s + 4.35e5·29-s − 2.92e4·31-s − 5.42e3·32-s + 2.57e5·34-s + 2.46e5·35-s − 7.09e5·37-s − 5.73e5·38-s + 6.12e5·40-s + 2.50e4·41-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 0.554·4-s + 1.28·5-s + 0.755·7-s + 1.17·8-s + 1.02·10-s + 1.11·11-s + 0.973·13-s + 0.601·14-s + 0.523·16-s + 1.41·17-s − 2.13·19-s + 0.714·20-s + 0.888·22-s − 1.40·23-s + 0.607·25-s + 0.774·26-s + 0.419·28-s + 3.31·29-s − 0.176·31-s − 0.0292·32-s + 1.12·34-s + 0.973·35-s − 2.30·37-s − 1.69·38-s + 1.51·40-s + 0.0567·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.670890634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.670890634\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 9 T + 5 p T^{2} - 9 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 72 p T + 3286 p^{2} T^{2} - 72 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4932 T + 19797958 T^{2} - 4932 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7708 T + 118266510 T^{2} - 7708 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 28584 T + 942631150 T^{2} - 28584 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 63728 T + 2569797414 T^{2} + 63728 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82260 T + 8202169294 T^{2} + 82260 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 435996 T + 80862098782 T^{2} - 435996 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 29240 T + 27798421182 T^{2} + 29240 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 709556 T + 313296440190 T^{2} + 709556 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 25056 T - 26592839954 T^{2} - 25056 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 496216 T + 567160908438 T^{2} - 496216 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1575000 T + 1564490669086 T^{2} - 1575000 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2057436 T + 3149566808638 T^{2} + 2057436 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1101024 T + 4521593481622 T^{2} - 1101024 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 28996 T + 5887677435486 T^{2} - 28996 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4480784 T + 16777543143750 T^{2} + 4480784 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 54540 T + 17803030672942 T^{2} + 54540 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 666604 T - 310686963642 T^{2} - 666604 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2322952 T + 38986658128734 T^{2} - 2322952 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7384392 T + 61089776413510 T^{2} - 7384392 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1784448 T + 26626978018894 T^{2} + 1784448 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16266412 T + 223770781220502 T^{2} - 16266412 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−13.73780031003017593127874337547, −13.65933621907489504632711700536, −12.44725265615688808248449363109, −12.37297534933374059084111276765, −11.69725532208648467700232431689, −10.70830530010260655380968437856, −10.51547012616072731684136090074, −10.06364704050250353079918100319, −8.985628912979882489433717432548, −8.556125760460655857240664738784, −7.85203233050783422061855791682, −6.89979177745247035948504947537, −6.20360516658398471502640467409, −5.95321818253614212945680652922, −4.91242463056994085717255765124, −4.34655073920729474086812870920, −3.61043453502218327639942614976, −2.31997476284253940317969688819, −1.70161263079764655666835009511, −1.05152655414564401617120043169,
1.05152655414564401617120043169, 1.70161263079764655666835009511, 2.31997476284253940317969688819, 3.61043453502218327639942614976, 4.34655073920729474086812870920, 4.91242463056994085717255765124, 5.95321818253614212945680652922, 6.20360516658398471502640467409, 6.89979177745247035948504947537, 7.85203233050783422061855791682, 8.556125760460655857240664738784, 8.985628912979882489433717432548, 10.06364704050250353079918100319, 10.51547012616072731684136090074, 10.70830530010260655380968437856, 11.69725532208648467700232431689, 12.37297534933374059084111276765, 12.44725265615688808248449363109, 13.65933621907489504632711700536, 13.73780031003017593127874337547
