| L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 3·6-s + 10·8-s − 11-s − 6·12-s + 3·13-s + 15·16-s − 2·17-s + 3·19-s − 3·22-s − 2·23-s − 10·24-s − 25-s + 9·26-s − 2·29-s + 22·32-s + 33-s − 6·34-s + 9·38-s − 3·39-s − 6·44-s − 6·46-s − 15·48-s − 49-s − 3·50-s + ⋯ |
| L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 3·6-s + 10·8-s − 11-s − 6·12-s + 3·13-s + 15·16-s − 2·17-s + 3·19-s − 3·22-s − 2·23-s − 10·24-s − 25-s + 9·26-s − 2·29-s + 22·32-s + 33-s − 6·34-s + 9·38-s − 3·39-s − 6·44-s − 6·46-s − 15·48-s − 49-s − 3·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(8.691883791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.691883791\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−6.43063132910694996140041416001, −6.33566622316266720514617648492, −6.20159755150888266028503701119, −5.81820661742036333616827116408, −5.78026527202102135929968583138, −5.59971976181835431385930481035, −5.56919691033676277124904450063, −5.22468041487568057483296394102, −5.19702476317934490884801871754, −5.01299527421013749686078410533, −4.29358346710920206862928083509, −4.28606516916897942383683105997, −4.11513672882863889336920846109, −3.95289697488769425724936961715, −3.89692965906837451583909724345, −3.59221123046221468534299891148, −3.24588558521075082405944708887, −2.87667747755197645752423453914, −2.75485978529463267966960097434, −2.64410846358512875662186522678, −2.25098831899809015604863774115, −1.68886661510674863768926731397, −1.57544670175381383041692720923, −1.51168696704987709529697247973, −0.902967577569815416861286500863,
0.902967577569815416861286500863, 1.51168696704987709529697247973, 1.57544670175381383041692720923, 1.68886661510674863768926731397, 2.25098831899809015604863774115, 2.64410846358512875662186522678, 2.75485978529463267966960097434, 2.87667747755197645752423453914, 3.24588558521075082405944708887, 3.59221123046221468534299891148, 3.89692965906837451583909724345, 3.95289697488769425724936961715, 4.11513672882863889336920846109, 4.28606516916897942383683105997, 4.29358346710920206862928083509, 5.01299527421013749686078410533, 5.19702476317934490884801871754, 5.22468041487568057483296394102, 5.56919691033676277124904450063, 5.59971976181835431385930481035, 5.78026527202102135929968583138, 5.81820661742036333616827116408, 6.20159755150888266028503701119, 6.33566622316266720514617648492, 6.43063132910694996140041416001
