| L(s) = 1 | − 3·4-s + 41·16-s + 216·17-s + 120·23-s − 228·25-s + 48·29-s − 1.06e3·43-s − 328·49-s + 864·53-s − 2.28e3·61-s − 411·64-s − 648·68-s + 288·79-s − 360·92-s + 684·100-s + 528·101-s + 3.24e3·107-s − 1.41e3·113-s − 144·116-s − 4.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3/8·4-s + 0.640·16-s + 3.08·17-s + 1.08·23-s − 1.82·25-s + 0.307·29-s − 3.77·43-s − 0.956·49-s + 2.23·53-s − 4.78·61-s − 0.802·64-s − 1.15·68-s + 0.410·79-s − 0.407·92-s + 0.683·100-s + 0.520·101-s + 2.92·107-s − 1.17·113-s − 0.115·116-s − 3.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9864211708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9864211708\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 T^{2} - p^{5} T^{4} + 3 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 228 T^{2} + 33862 T^{4} + 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 328 T^{2} + 51918 T^{4} + 328 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4896 T^{2} + 9533230 T^{4} + 4896 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 15880 T^{2} + 155242878 T^{4} + 15880 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 60 T + 1870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 25558 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 102136 T^{2} + 4357504590 T^{4} + 102136 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 2164 T^{2} - 2618990922 T^{4} + 2164 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 230292 T^{2} + 22372016182 T^{4} + 230292 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 532 T + 206406 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 85056 T^{2} + 23167872958 T^{4} - 85056 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 432 T + 134134 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 716592 T^{2} + 212420895502 T^{4} + 716592 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 1140 T + 12598 p T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 475384 T^{2} + 105182398878 T^{4} + 475384 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 1283952 T^{2} + 666179277502 T^{4} + 1283952 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 762916 T^{2} + 359882924838 T^{4} + 762916 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 144 T + 825118 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 1633872 T^{2} + 191792062 p^{2} T^{4} + 1633872 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 2759412 T^{2} + 2896886558902 T^{4} + 2759412 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 53564 T^{2} + 1619815156998 T^{4} - 53564 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.37930659496544574397705670879, −6.29721064039630347016816338798, −5.75838264734123836280183254451, −5.70776736150244280323402271041, −5.50181541502453252223599188487, −5.33535249208491691802886796430, −5.08633098353777881636386391772, −4.79053798433035189212045160954, −4.76605052051313146434766120682, −4.33345728255767385996132094379, −4.11462068124171021169062391522, −3.91861308170087015614107261167, −3.42519848662332914634031728735, −3.34162545487194498416355188288, −3.23048150081354990052309841992, −3.10532885501930015910677477839, −2.81039842989896806071099652258, −2.30695114666928373366558088683, −1.93389277118341985675288936185, −1.67920503356369532511487561673, −1.51764927910245479210637360112, −1.20828584253536194609831584715, −0.864712926976473255208328020719, −0.56706627024054596407793270340, −0.11453834143139711145686727803,
0.11453834143139711145686727803, 0.56706627024054596407793270340, 0.864712926976473255208328020719, 1.20828584253536194609831584715, 1.51764927910245479210637360112, 1.67920503356369532511487561673, 1.93389277118341985675288936185, 2.30695114666928373366558088683, 2.81039842989896806071099652258, 3.10532885501930015910677477839, 3.23048150081354990052309841992, 3.34162545487194498416355188288, 3.42519848662332914634031728735, 3.91861308170087015614107261167, 4.11462068124171021169062391522, 4.33345728255767385996132094379, 4.76605052051313146434766120682, 4.79053798433035189212045160954, 5.08633098353777881636386391772, 5.33535249208491691802886796430, 5.50181541502453252223599188487, 5.70776736150244280323402271041, 5.75838264734123836280183254451, 6.29721064039630347016816338798, 6.37930659496544574397705670879
