| L(s) = 1 | − 3·5-s − 20·7-s − 51·11-s + 122·13-s + 24·17-s − 169·19-s − 192·23-s + 124·25-s + 78·29-s + 92·31-s + 60·35-s + 173·37-s − 348·41-s − 994·43-s + 180·47-s + 127·49-s + 285·53-s + 153·55-s − 1.26e3·59-s − 328·61-s − 366·65-s + 875·67-s + 2.80e3·71-s + 1.36e3·73-s + 1.02e3·77-s + 182·79-s + 798·83-s + ⋯ |
| L(s) = 1 | − 0.268·5-s − 1.07·7-s − 1.39·11-s + 2.60·13-s + 0.342·17-s − 2.04·19-s − 1.74·23-s + 0.991·25-s + 0.499·29-s + 0.533·31-s + 0.289·35-s + 0.768·37-s − 1.32·41-s − 3.52·43-s + 0.558·47-s + 0.370·49-s + 0.738·53-s + 0.375·55-s − 2.80·59-s − 0.688·61-s − 0.698·65-s + 1.59·67-s + 4.69·71-s + 2.18·73-s + 1.50·77-s + 0.259·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1657542178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1657542178\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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 | $D_{4}$ | \( ( 1 - 841 T + 1945608 T^{2} - 841 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 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
−8.339931259615206987548378838299, −8.126736002717176302962911015226, −7.984195033317557187493964641075, −7.80675738459040371438924046372, −7.07119791041385227407558123198, −6.80830707133902763410965136449, −6.67541479118251028953386166663, −6.37804541271960320505743821693, −6.34372779517028820917989802959, −5.88650312182277215613492760484, −5.80870439502315264124942488377, −5.13693794376654035208045225140, −5.03666387040549919646068969394, −4.78743006268608402040249205001, −4.28055608045716457502284919448, −3.96416539822846191491645940977, −3.48945111338199670147891375548, −3.41427486648488404713490255534, −3.33795141277712125428586543473, −2.55433378406029226241405310717, −2.15766497628013387519010378396, −2.01257568841959446359309684880, −1.23197585453533349326073923669, −0.823594089859419767699424459504, −0.087113658155435372054388344910,
0.087113658155435372054388344910, 0.823594089859419767699424459504, 1.23197585453533349326073923669, 2.01257568841959446359309684880, 2.15766497628013387519010378396, 2.55433378406029226241405310717, 3.33795141277712125428586543473, 3.41427486648488404713490255534, 3.48945111338199670147891375548, 3.96416539822846191491645940977, 4.28055608045716457502284919448, 4.78743006268608402040249205001, 5.03666387040549919646068969394, 5.13693794376654035208045225140, 5.80870439502315264124942488377, 5.88650312182277215613492760484, 6.34372779517028820917989802959, 6.37804541271960320505743821693, 6.67541479118251028953386166663, 6.80830707133902763410965136449, 7.07119791041385227407558123198, 7.80675738459040371438924046372, 7.984195033317557187493964641075, 8.126736002717176302962911015226, 8.339931259615206987548378838299
