| L(s) = 1 | − 2·2-s + 6·3-s + 2·4-s + 8·5-s − 12·6-s − 12·7-s − 4·8-s + 16·9-s − 16·10-s + 4·11-s + 12·12-s + 24·14-s + 48·15-s + 8·16-s − 6·17-s − 32·18-s + 6·19-s + 16·20-s − 72·21-s − 8·22-s − 6·23-s − 24·24-s + 26·25-s + 24·27-s − 24·28-s − 96·30-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3.46·3-s + 4-s + 3.57·5-s − 4.89·6-s − 4.53·7-s − 1.41·8-s + 16/3·9-s − 5.05·10-s + 1.20·11-s + 3.46·12-s + 6.41·14-s + 12.3·15-s + 2·16-s − 1.45·17-s − 7.54·18-s + 1.37·19-s + 3.57·20-s − 15.7·21-s − 1.70·22-s − 1.25·23-s − 4.89·24-s + 26/5·25-s + 4.61·27-s − 4.53·28-s − 17.5·30-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485486766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485486766\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T + 20 T^{2} - 16 p T^{3} + 91 T^{4} - 16 p^{2} T^{5} + 20 p^{2} T^{6} - 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1674 T^{3} + 11396 T^{4} + 1674 p T^{5} + 201 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 110 T^{2} + 744 T^{3} + 4059 T^{4} + 744 p T^{5} + 110 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 6291 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T - 58 T^{2} - 32 T^{3} + 7627 T^{4} - 32 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 1404 p T^{5} + 112 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 144 T^{3} - 3613 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 158 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 2088 T^{3} + 26627 T^{4} - 2088 p T^{5} + 222 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03075847834763098474882288151, −9.702696483346020277062097088550, −9.512756683316004384601837538598, −9.387903522833939650182598234837, −9.210672170272203953931233207164, −8.908559251459995551990744801681, −8.834357663926325852606602758190, −8.408860199936460454095466567285, −8.312494517281831210401192353994, −7.70288829964671435338272721127, −7.10000603151769069608768436595, −7.00963515152245142485097021304, −6.46623293715454287649065243745, −6.30479423701509434950843452068, −6.14559647065851798915674748456, −6.09494337762618281372472474993, −5.55034809440366847308401048876, −4.88613071800989277155250540128, −3.60904537023455496062226159296, −3.52620897108498128919421212588, −3.50044679182669776362616364966, −2.79563494400777199708628505245, −2.67635887257639481241164146880, −2.22246397514802082514205269828, −1.84456934600409734569245529865,
1.84456934600409734569245529865, 2.22246397514802082514205269828, 2.67635887257639481241164146880, 2.79563494400777199708628505245, 3.50044679182669776362616364966, 3.52620897108498128919421212588, 3.60904537023455496062226159296, 4.88613071800989277155250540128, 5.55034809440366847308401048876, 6.09494337762618281372472474993, 6.14559647065851798915674748456, 6.30479423701509434950843452068, 6.46623293715454287649065243745, 7.00963515152245142485097021304, 7.10000603151769069608768436595, 7.70288829964671435338272721127, 8.312494517281831210401192353994, 8.408860199936460454095466567285, 8.834357663926325852606602758190, 8.908559251459995551990744801681, 9.210672170272203953931233207164, 9.387903522833939650182598234837, 9.512756683316004384601837538598, 9.702696483346020277062097088550, 10.03075847834763098474882288151
