| L(s) = 1 | + 8·11-s + 8·17-s + 8·19-s − 8·41-s − 24·43-s − 4·49-s − 48·67-s − 24·73-s − 8·83-s + 40·89-s − 16·97-s + 32·107-s + 8·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 1.94·17-s + 1.83·19-s − 1.24·41-s − 3.65·43-s − 4/7·49-s − 5.86·67-s − 2.80·73-s − 0.878·83-s + 4.23·89-s − 1.62·97-s + 3.09·107-s + 0.752·113-s + 3.27·121-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 − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.785348205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785348205\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 114 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 48 T^{2} + 3314 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 128 T^{2} + 8658 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 + 188 T^{2} + 17638 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 260 T^{2} + 28662 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.54853818601124618285808851202, −4.96449285221630790872450106688, −4.96336526177909387919467491295, −4.90146571164645180681075030367, −4.81620156760291470860111113950, −4.63032362931823206459374665275, −4.28565788096700233173846592310, −4.02870764381218048278446932328, −3.96690612812597804221730770381, −3.56820328196675873658085947440, −3.47673582071718593594970157142, −3.35585855790566381604924758455, −3.24091933794327148899372516809, −3.09280680790657636149640816684, −2.92045410665745938337660810615, −2.46883555045939784608834966428, −2.44549259533023530298365177468, −1.71906008566740756667303058970, −1.68135219288911631158372933319, −1.60031086907669443874014908631, −1.54350466393392444657537822677, −1.10322537785739001405192425345, −1.02589891298707648236413125191, −0.58767304050968198216843048996, −0.13679966775258260570584844736,
0.13679966775258260570584844736, 0.58767304050968198216843048996, 1.02589891298707648236413125191, 1.10322537785739001405192425345, 1.54350466393392444657537822677, 1.60031086907669443874014908631, 1.68135219288911631158372933319, 1.71906008566740756667303058970, 2.44549259533023530298365177468, 2.46883555045939784608834966428, 2.92045410665745938337660810615, 3.09280680790657636149640816684, 3.24091933794327148899372516809, 3.35585855790566381604924758455, 3.47673582071718593594970157142, 3.56820328196675873658085947440, 3.96690612812597804221730770381, 4.02870764381218048278446932328, 4.28565788096700233173846592310, 4.63032362931823206459374665275, 4.81620156760291470860111113950, 4.90146571164645180681075030367, 4.96336526177909387919467491295, 4.96449285221630790872450106688, 5.54853818601124618285808851202
