| L(s) = 1 | − 9·2-s + 54·3-s + 71·4-s + 360·5-s − 486·6-s − 1.70e3·8-s + 2.18e3·9-s − 3.24e3·10-s − 4.93e3·11-s + 3.83e3·12-s − 7.70e3·13-s + 1.94e4·15-s + 8.58e3·16-s + 2.85e4·17-s − 1.96e4·18-s + 6.37e4·19-s + 2.55e4·20-s + 4.43e4·22-s + 8.22e4·23-s − 9.18e4·24-s + 4.74e4·25-s + 6.93e4·26-s + 7.87e4·27-s − 4.35e5·29-s − 1.74e5·30-s + 2.92e4·31-s + 5.42e3·32-s + ⋯ |
| L(s) = 1 | − 0.795·2-s + 1.15·3-s + 0.554·4-s + 1.28·5-s − 0.918·6-s − 1.17·8-s + 9-s − 1.02·10-s − 1.11·11-s + 0.640·12-s − 0.973·13-s + 1.48·15-s + 0.523·16-s + 1.41·17-s − 0.795·18-s + 2.13·19-s + 0.714·20-s + 0.888·22-s + 1.40·23-s − 1.35·24-s + 0.607·25-s + 0.774·26-s + 0.769·27-s − 3.31·29-s − 1.18·30-s + 0.176·31-s + 0.0292·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.394525638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.394525638\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| 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
−12.17194808505181664851836756233, −11.40740754381307548733309817841, −10.64636041175394145263922036854, −10.26166734219820032173150616518, −9.702836448443530529533169478943, −9.366170832393655326051916632193, −9.104288339023189733146548534532, −8.552153631033073184073740374082, −7.62469138142830954311097062862, −7.27051018811373868993735188968, −7.20781955254161867796138624701, −5.87440838188882320081383545989, −5.33581155736152869433806119409, −5.30491730594928360800060387478, −3.76439390611912330007713611104, −3.14153424586338509323792174240, −2.62276532865152745559808205230, −2.09821564960457890131335357412, −1.33432434089120765658982273189, −0.53209085803341718743993492250,
0.53209085803341718743993492250, 1.33432434089120765658982273189, 2.09821564960457890131335357412, 2.62276532865152745559808205230, 3.14153424586338509323792174240, 3.76439390611912330007713611104, 5.30491730594928360800060387478, 5.33581155736152869433806119409, 5.87440838188882320081383545989, 7.20781955254161867796138624701, 7.27051018811373868993735188968, 7.62469138142830954311097062862, 8.552153631033073184073740374082, 9.104288339023189733146548534532, 9.366170832393655326051916632193, 9.702836448443530529533169478943, 10.26166734219820032173150616518, 10.64636041175394145263922036854, 11.40740754381307548733309817841, 12.17194808505181664851836756233
