| L(s) = 1 | − 2·9-s + 2·11-s − 10·19-s + 24·29-s − 8·31-s − 20·41-s − 2·49-s + 16·59-s + 24·61-s − 16·71-s + 28·79-s + 9·81-s + 12·89-s − 4·99-s − 24·101-s − 8·109-s + 23·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 0.603·11-s − 2.29·19-s + 4.45·29-s − 1.43·31-s − 3.12·41-s − 2/7·49-s + 2.08·59-s + 3.07·61-s − 1.89·71-s + 3.15·79-s + 81-s + 1.27·89-s − 0.402·99-s − 2.38·101-s − 0.766·109-s + 2.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.736560605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736560605\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 15 T^{2} - 2584 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 2 T^{2} - 5325 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.82708818289721519086904105408, −6.56597762052751732988513436027, −6.47887112433296232073742489007, −6.35562038286817850766142056728, −5.96972413819591610267129714264, −5.70032763215459659024571135303, −5.58670142686039896710079978726, −5.17153634662819944737670411144, −4.99809023060385495247102344943, −4.96541716030442589235518553448, −4.48728692404637249036578006919, −4.44006574932971259656464994554, −4.17887144443436498109158338451, −3.78452168032819384951494684738, −3.58395363244916475216193540501, −3.39322805017192450587754902124, −3.21309133647609429473080002735, −2.63836008266889495588638296476, −2.51435494090858670129710842555, −2.33219141980804770340483815664, −1.97601280831452375196762100091, −1.62329028416028736711852175576, −1.25058585158651339437805787567, −0.68876421221699789902845641128, −0.43085976744882913828925056628,
0.43085976744882913828925056628, 0.68876421221699789902845641128, 1.25058585158651339437805787567, 1.62329028416028736711852175576, 1.97601280831452375196762100091, 2.33219141980804770340483815664, 2.51435494090858670129710842555, 2.63836008266889495588638296476, 3.21309133647609429473080002735, 3.39322805017192450587754902124, 3.58395363244916475216193540501, 3.78452168032819384951494684738, 4.17887144443436498109158338451, 4.44006574932971259656464994554, 4.48728692404637249036578006919, 4.96541716030442589235518553448, 4.99809023060385495247102344943, 5.17153634662819944737670411144, 5.58670142686039896710079978726, 5.70032763215459659024571135303, 5.96972413819591610267129714264, 6.35562038286817850766142056728, 6.47887112433296232073742489007, 6.56597762052751732988513436027, 6.82708818289721519086904105408
