| L(s) = 1 | − 6·3-s + 3·5-s + 20·7-s + 9·9-s − 51·11-s + 122·13-s − 18·15-s − 24·17-s + 169·19-s − 120·21-s − 192·23-s + 124·25-s + 54·27-s − 78·29-s − 92·31-s + 306·33-s + 60·35-s + 173·37-s − 732·39-s + 348·41-s + 994·43-s + 27·45-s + 180·47-s + 127·49-s + 144·51-s − 285·53-s − 153·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.268·5-s + 1.07·7-s + 1/3·9-s − 1.39·11-s + 2.60·13-s − 0.309·15-s − 0.342·17-s + 2.04·19-s − 1.24·21-s − 1.74·23-s + 0.991·25-s + 0.384·27-s − 0.499·29-s − 0.533·31-s + 1.61·33-s + 0.289·35-s + 0.768·37-s − 3.00·39-s + 1.32·41-s + 3.52·43-s + 0.0894·45-s + 0.558·47-s + 0.370·49-s + 0.395·51-s − 0.738·53-s − 0.375·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.568600628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.568600628\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 20 T + 39 p T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 3 T - 23 p T^{2} + 378 T^{3} - 1374 T^{4} + 378 p^{3} T^{5} - 23 p^{7} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 51 T - 53 p T^{2} + 26622 T^{3} + 4904364 T^{4} + 26622 p^{3} T^{5} - 53 p^{7} T^{6} + 51 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 61 T + 2118 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T - 1186 T^{2} - 193536 T^{3} - 23862813 T^{4} - 193536 p^{3} T^{5} - 1186 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 169 T + 7831 T^{2} - 1185028 T^{3} + 186787120 T^{4} - 1185028 p^{3} T^{5} + 7831 p^{6} T^{6} - 169 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 96 T - 2951 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 39 T + 12094 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 92 T - 40409 T^{2} - 985228 T^{3} + 1248915424 T^{4} - 985228 p^{3} T^{5} - 40409 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 173 T - 32561 T^{2} + 6715168 T^{3} - 176719946 T^{4} + 6715168 p^{3} T^{5} - 32561 p^{6} T^{6} - 173 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 174 T - 2846 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 497 T + 174468 T^{2} - 497 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 180 T - 150514 T^{2} + 4451760 T^{3} + 19314450867 T^{4} + 4451760 p^{3} T^{5} - 150514 p^{6} T^{6} - 180 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 285 T - 143341 T^{2} - 20858580 T^{3} + 16172992902 T^{4} - 20858580 p^{3} T^{5} - 143341 p^{6} T^{6} + 285 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1269 T + 798167 T^{2} + 8634276 p T^{3} + 82374096 p^{2} T^{4} + 8634276 p^{4} T^{5} + 798167 p^{6} T^{6} + 1269 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 328 T - 207062 T^{2} - 45695648 T^{3} + 23062207051 T^{4} - 45695648 p^{3} T^{5} - 207062 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 875 T - 16919 T^{2} + 158390750 T^{3} + 291645057892 T^{4} + 158390750 p^{3} T^{5} - 16919 p^{6} T^{6} + 875 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1404 T + 1077298 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1361 T + 667765 T^{2} - 553276442 T^{3} + 531254332390 T^{4} - 553276442 p^{3} T^{5} + 667765 p^{6} T^{6} - 1361 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 182 T - 959183 T^{2} + 1133678 T^{3} + 725254302892 T^{4} + 1133678 p^{3} T^{5} - 959183 p^{6} T^{6} + 182 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 399 T + 946240 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 822 T - 841102 T^{2} + 87829056 T^{3} + 1327322205039 T^{4} + 87829056 p^{3} T^{5} - 841102 p^{6} T^{6} + 822 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 841 T + 1945608 T^{2} - 841 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.83276136537124684542835303103, −7.55882750348679987614215356424, −7.55553637373397120580651562594, −7.54239058197364302563797606759, −6.71007503630590430637969112669, −6.47971324926329654445316712235, −6.36420146142533051544538610071, −5.99424193187340429351892347169, −5.83662219045582266930403433230, −5.69485935642311279206395016642, −5.23029389241646082334502321492, −5.17701630209141828746995074457, −4.92820625694919743094041158263, −4.36027470163902780179713220034, −4.21380850001322398727301244247, −3.96421730892818759316881990939, −3.38671547859138471078503852012, −3.34339723972194938524730313630, −2.62454143272499928125932940399, −2.57059613881110536750646480577, −1.87237203208634086000365996236, −1.68281005928335084044082621519, −0.977613465305263028358899522633, −0.847399117709408841325485913908, −0.47729186977758951895913446369,
0.47729186977758951895913446369, 0.847399117709408841325485913908, 0.977613465305263028358899522633, 1.68281005928335084044082621519, 1.87237203208634086000365996236, 2.57059613881110536750646480577, 2.62454143272499928125932940399, 3.34339723972194938524730313630, 3.38671547859138471078503852012, 3.96421730892818759316881990939, 4.21380850001322398727301244247, 4.36027470163902780179713220034, 4.92820625694919743094041158263, 5.17701630209141828746995074457, 5.23029389241646082334502321492, 5.69485935642311279206395016642, 5.83662219045582266930403433230, 5.99424193187340429351892347169, 6.36420146142533051544538610071, 6.47971324926329654445316712235, 6.71007503630590430637969112669, 7.54239058197364302563797606759, 7.55553637373397120580651562594, 7.55882750348679987614215356424, 7.83276136537124684542835303103
