| L(s) = 1 | + 92·4-s + 598·7-s + 4.99e3·13-s + 4.36e3·16-s − 5.01e3·19-s − 2.63e4·25-s + 5.50e4·28-s + 1.06e4·31-s + 6.51e4·37-s − 1.41e5·43-s + 3.29e4·49-s + 4.59e5·52-s − 1.23e5·61-s + 2.50e4·64-s − 8.60e5·67-s + 5.03e5·73-s − 4.61e5·76-s + 1.32e6·79-s + 2.98e6·91-s + 4.41e5·97-s − 2.42e6·100-s − 1.18e6·103-s + 3.02e6·109-s + 2.61e6·112-s + 3.15e6·121-s + 9.80e5·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.43·4-s + 1.74·7-s + 2.27·13-s + 1.06·16-s − 0.731·19-s − 1.68·25-s + 2.50·28-s + 0.357·31-s + 1.28·37-s − 1.77·43-s + 0.279·49-s + 3.26·52-s − 0.544·61-s + 0.0954·64-s − 2.86·67-s + 1.29·73-s − 1.05·76-s + 2.68·79-s + 3.95·91-s + 0.483·97-s − 2.42·100-s − 1.08·103-s + 2.33·109-s + 1.85·112-s + 1.78·121-s + 0.514·124-s − 1.27·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.880602172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.880602172\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 23 p^{2} T^{2} + p^{12} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 14 p T + p^{6} T^{2} )( 1 + 14 p T + p^{6} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 299 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3153746 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2495 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 44770754 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2509 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 170306 p^{2} T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 627387698 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5330 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 32591 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5125179746 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 70630 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21542558402 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7869111122 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 28021297682 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 61801 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 430261 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 192841636898 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 251615 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 660827 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 17306550002 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 920751210146 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 220727 T + p^{6} T^{2} )^{2} \) |
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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
−16.32901971440025001174604379715, −15.67967349470792618529683367992, −15.05649146261494812633409485461, −14.83552838579428360049450296384, −13.60520871549574784535648973829, −13.57293222126820040829193728746, −12.29168464788704310873779048123, −11.62369540682971623391568410023, −11.05625979805148337460201676372, −11.01828444618274095997147346912, −9.990486139180967140208432314978, −8.733367916535631938827728697853, −8.111968754919570520732082179980, −7.60224623730087719735163842154, −6.36916925635655843099364052175, −6.00095655174430841648749994050, −4.68841056120705700074402565526, −3.53858300520273510886731839187, −2.03232292531174936133167360743, −1.37340758254179577886894900599,
1.37340758254179577886894900599, 2.03232292531174936133167360743, 3.53858300520273510886731839187, 4.68841056120705700074402565526, 6.00095655174430841648749994050, 6.36916925635655843099364052175, 7.60224623730087719735163842154, 8.111968754919570520732082179980, 8.733367916535631938827728697853, 9.990486139180967140208432314978, 11.01828444618274095997147346912, 11.05625979805148337460201676372, 11.62369540682971623391568410023, 12.29168464788704310873779048123, 13.57293222126820040829193728746, 13.60520871549574784535648973829, 14.83552838579428360049450296384, 15.05649146261494812633409485461, 15.67967349470792618529683367992, 16.32901971440025001174604379715
