| L(s) = 1 | − 3·5-s − 6·17-s − 6·19-s − 30·23-s + 9·25-s − 16·31-s + 48·47-s + 137·49-s + 192·53-s + 38·61-s + 6·79-s − 288·83-s + 18·85-s + 18·95-s + 18·107-s − 226·109-s + 564·113-s + 90·115-s + 129·121-s − 102·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + ⋯ |
| L(s) = 1 | − 3/5·5-s − 0.352·17-s − 0.315·19-s − 1.30·23-s + 9/25·25-s − 0.516·31-s + 1.02·47-s + 2.79·49-s + 3.62·53-s + 0.622·61-s + 6/79·79-s − 3.46·83-s + 0.211·85-s + 0.189·95-s + 0.168·107-s − 2.07·109-s + 4.99·113-s + 0.782·115-s + 1.06·121-s − 0.815·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.309·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.888538046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888538046\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 137 T^{2} + 9024 T^{4} - 137 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 129 T^{2} + 10400 T^{4} - 129 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 136 T^{2} - 5970 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 528 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 3 T + 254 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 15 T + 1062 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1522 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3186 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 776 T^{2} + 5716590 T^{4} - 776 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 24 T + 3726 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 96 T + 6041 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7044 T^{2} + 28934630 T^{4} - 7044 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 19 T + 7062 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6584 T^{2} + 19350606 T^{4} - 6584 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 3 T + 8252 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 144 T + 13737 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1879 T^{2} + 54186000 T^{4} + 1879 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.45275063207953615304089024260, −5.79164533082923924781390514747, −5.79008241202683818535708973968, −5.74215812804999647912029325241, −5.48222760952389867665591201241, −5.24645140823958225849295368585, −5.11701133734558483652206175100, −4.68241730709267918121942179489, −4.33827998672569627841733873404, −4.33426453395391530260449804978, −4.12483819882266062802836670877, −3.83751839180455988459629681127, −3.77733451259936906185745775943, −3.62249088879339623771724542396, −3.01140912643776800838049716719, −2.92262854989699602723186920788, −2.60865609810213438551997379665, −2.48003195240326275318872101856, −2.11813636678865547521278669505, −1.84655260032679177117511499079, −1.69763018371610120969033403761, −1.07857019234062449529969872411, −0.880902383580385417073733515747, −0.55606554905684952155676777865, −0.25989186106014136577233636213,
0.25989186106014136577233636213, 0.55606554905684952155676777865, 0.880902383580385417073733515747, 1.07857019234062449529969872411, 1.69763018371610120969033403761, 1.84655260032679177117511499079, 2.11813636678865547521278669505, 2.48003195240326275318872101856, 2.60865609810213438551997379665, 2.92262854989699602723186920788, 3.01140912643776800838049716719, 3.62249088879339623771724542396, 3.77733451259936906185745775943, 3.83751839180455988459629681127, 4.12483819882266062802836670877, 4.33426453395391530260449804978, 4.33827998672569627841733873404, 4.68241730709267918121942179489, 5.11701133734558483652206175100, 5.24645140823958225849295368585, 5.48222760952389867665591201241, 5.74215812804999647912029325241, 5.79008241202683818535708973968, 5.79164533082923924781390514747, 6.45275063207953615304089024260
