| L(s) = 1 | − 6·3-s − 3·5-s + 9·9-s + 51·11-s − 122·13-s + 18·15-s + 24·17-s + 169·19-s + 192·23-s + 124·25-s + 54·27-s − 78·29-s − 92·31-s − 306·33-s + 173·37-s + 732·39-s − 348·41-s − 994·43-s − 27·45-s + 180·47-s − 144·51-s − 285·53-s − 153·55-s − 1.01e3·57-s − 1.26e3·59-s + 328·61-s + 366·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.268·5-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 + 1.74·23-s + 0.991·25-s + 0.384·27-s − 0.499·29-s − 0.533·31-s − 1.61·33-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 − 0.395·51-s − 0.738·53-s − 0.375·55-s − 2.35·57-s − 2.80·59-s + 0.688·61-s + 0.698·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2740018702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2740018702\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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 | $D_{4}$ | \( ( 1 + 841 T + 1945608 T^{2} + 841 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.27054161872513760086050775764, −7.10719031894088817988546950690, −6.69123885743128648219891912420, −6.63683367514686818768926234791, −6.46607861742881526716299339979, −6.18697242892464661326049025264, −5.67450165108154360326489512651, −5.38821474924641140247003884110, −5.24234120158685951583743009332, −5.11280721387812438938278068994, −5.10302214380054221955049077615, −4.62057047159707113028859450741, −4.29355258273341553393091532460, −4.09571843845541839398790917037, −3.88070649834038355842515099179, −3.02887206235046284375428809999, −2.99711963941344517688970088082, −2.97860823145643375278612658508, −2.94650425930451893801797998430, −1.92018403740098361058566443051, −1.64741665556912293653378585974, −1.41679011685305680249997197603, −1.10642008696984249009701560596, −0.51518012410538863732201885457, −0.11178613706943458676915876835,
0.11178613706943458676915876835, 0.51518012410538863732201885457, 1.10642008696984249009701560596, 1.41679011685305680249997197603, 1.64741665556912293653378585974, 1.92018403740098361058566443051, 2.94650425930451893801797998430, 2.97860823145643375278612658508, 2.99711963941344517688970088082, 3.02887206235046284375428809999, 3.88070649834038355842515099179, 4.09571843845541839398790917037, 4.29355258273341553393091532460, 4.62057047159707113028859450741, 5.10302214380054221955049077615, 5.11280721387812438938278068994, 5.24234120158685951583743009332, 5.38821474924641140247003884110, 5.67450165108154360326489512651, 6.18697242892464661326049025264, 6.46607861742881526716299339979, 6.63683367514686818768926234791, 6.69123885743128648219891912420, 7.10719031894088817988546950690, 7.27054161872513760086050775764
