| L(s) = 1 | − 4·5-s + 9-s + 2·13-s − 2·17-s + 6·25-s + 2·29-s + 2·37-s − 2·41-s − 4·45-s + 49-s + 2·53-s − 8·65-s − 2·73-s + 8·85-s − 2·89-s + 2·97-s + 4·101-s + 2·109-s + 2·113-s + 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + ⋯ |
| L(s) = 1 | − 4·5-s + 9-s + 2·13-s − 2·17-s + 6·25-s + 2·29-s + 2·37-s − 2·41-s − 4·45-s + 49-s + 2·53-s − 8·65-s − 2·73-s + 8·85-s − 2·89-s + 2·97-s + 4·101-s + 2·109-s + 2·113-s + 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4016461478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4016461478\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 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
−7.36669278897958561540550653899, −7.29562196791550169204664183095, −6.98124885064642614002063132719, −6.74337051319561540021444648016, −6.70940938097809815351325426679, −6.17432296555585313667241817814, −6.03184782441006453297710064859, −5.94405003289741859636861953826, −5.68302501438110682787747212629, −5.06557388815782573449609739789, −4.86188107277528562920809441933, −4.54750616976340650115264709035, −4.35816064857474253969256452712, −4.35521183471516629242609588060, −4.02858587833784317557035126449, −4.01892293615591860649017238021, −3.45722881146812250629576536470, −3.45445309405558840351209183118, −3.25443275792910673354172227913, −2.89623552428398970464238200300, −2.28449149183528066827763165953, −2.09468939711511061014624055451, −1.64476090390049158851296281161, −0.924602959498739877767318876373, −0.68565665448265552785458784317,
0.68565665448265552785458784317, 0.924602959498739877767318876373, 1.64476090390049158851296281161, 2.09468939711511061014624055451, 2.28449149183528066827763165953, 2.89623552428398970464238200300, 3.25443275792910673354172227913, 3.45445309405558840351209183118, 3.45722881146812250629576536470, 4.01892293615591860649017238021, 4.02858587833784317557035126449, 4.35521183471516629242609588060, 4.35816064857474253969256452712, 4.54750616976340650115264709035, 4.86188107277528562920809441933, 5.06557388815782573449609739789, 5.68302501438110682787747212629, 5.94405003289741859636861953826, 6.03184782441006453297710064859, 6.17432296555585313667241817814, 6.70940938097809815351325426679, 6.74337051319561540021444648016, 6.98124885064642614002063132719, 7.29562196791550169204664183095, 7.36669278897958561540550653899
