| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·11-s + 2·12-s − 4·17-s + 4·19-s − 25-s − 2·27-s + 4·33-s + 36-s + 2·41-s + 2·43-s + 2·44-s − 8·51-s + 8·57-s − 64-s − 4·68-s + 4·73-s − 2·75-s + 4·76-s − 4·81-s + 2·83-s + 4·89-s − 2·97-s + 2·99-s − 100-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·11-s + 2·12-s − 4·17-s + 4·19-s − 25-s − 2·27-s + 4·33-s + 36-s + 2·41-s + 2·43-s + 2·44-s − 8·51-s + 8·57-s − 64-s − 4·68-s + 4·73-s − 2·75-s + 4·76-s − 4·81-s + 2·83-s + 4·89-s − 2·97-s + 2·99-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.497521197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.497521197\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−6.28848354183098440869960992510, −6.06797668180554012070437828358, −6.01955147909703378458623604447, −5.74655184159209436091131081136, −5.43454428395378089117626020205, −5.18273575528570436287194926112, −5.14912221210930446906175891689, −4.83530283288931935158073291681, −4.44356868339979147266704765835, −4.36576616958890310197621848064, −4.08412709092419099046998185087, −3.79309271843123663555992988523, −3.77697959285198129201053515895, −3.73996156370182854040237790774, −3.14884189605746295658297910567, −3.11891320957811421687197323999, −2.81795316954662185175246392405, −2.71452944613876158883694062888, −2.24039789460314081438766832399, −2.19774202771391952553253518933, −1.98869637236653478041155360697, −1.98552560539531269385561741501, −1.29134023109703930647668147695, −1.09722572691000892301421445637, −0.73442143665089481576984442712,
0.73442143665089481576984442712, 1.09722572691000892301421445637, 1.29134023109703930647668147695, 1.98552560539531269385561741501, 1.98869637236653478041155360697, 2.19774202771391952553253518933, 2.24039789460314081438766832399, 2.71452944613876158883694062888, 2.81795316954662185175246392405, 3.11891320957811421687197323999, 3.14884189605746295658297910567, 3.73996156370182854040237790774, 3.77697959285198129201053515895, 3.79309271843123663555992988523, 4.08412709092419099046998185087, 4.36576616958890310197621848064, 4.44356868339979147266704765835, 4.83530283288931935158073291681, 5.14912221210930446906175891689, 5.18273575528570436287194926112, 5.43454428395378089117626020205, 5.74655184159209436091131081136, 6.01955147909703378458623604447, 6.06797668180554012070437828358, 6.28848354183098440869960992510
