| L(s) = 1 | − 11·5-s − 5·11-s − 10·13-s − 100·17-s + 67·19-s + 76·23-s + 232·25-s − 550·29-s + 362·31-s + 5·37-s − 324·41-s + 1.44e3·43-s + 216·47-s + 495·53-s + 55·55-s − 173·59-s − 532·61-s + 110·65-s − 111·67-s − 3.20e3·71-s + 1.21e3·73-s − 1.46e3·79-s − 2.81e3·83-s + 1.10e3·85-s + 1.97e3·89-s − 737·95-s − 1.12e3·97-s + ⋯ |
| L(s) = 1 | − 0.983·5-s − 0.137·11-s − 0.213·13-s − 1.42·17-s + 0.808·19-s + 0.689·23-s + 1.85·25-s − 3.52·29-s + 2.09·31-s + 0.0222·37-s − 1.23·41-s + 5.11·43-s + 0.670·47-s + 1.28·53-s + 0.134·55-s − 0.381·59-s − 1.11·61-s + 0.209·65-s − 0.202·67-s − 5.34·71-s + 1.94·73-s − 2.07·79-s − 3.72·83-s + 1.40·85-s + 2.35·89-s − 0.795·95-s − 1.17·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.423726846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.423726846\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 11 T - 111 T^{2} - 198 T^{3} + 23074 T^{4} - 198 p^{3} T^{5} - 111 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 5 T - 279 T^{2} - 11790 T^{3} - 1712420 T^{4} - 11790 p^{3} T^{5} - 279 p^{6} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 3194 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 100 T - 1554 T^{2} + 172800 T^{3} + 60227347 T^{4} + 172800 p^{3} T^{5} - 1554 p^{6} T^{6} + 100 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 67 T - 9917 T^{2} - 46096 T^{3} + 129696904 T^{4} - 46096 p^{3} T^{5} - 9917 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 76 T - 702 T^{2} + 1357056 T^{3} - 176347997 T^{4} + 1357056 p^{3} T^{5} - 702 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 275 T + 61846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 362 T + 1871 p T^{2} - 4872882 T^{3} + 543844364 T^{4} - 4872882 p^{3} T^{5} + 1871 p^{7} T^{6} - 362 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 5 T - 3581 T^{2} + 488500 T^{3} - 2553989498 T^{4} + 488500 p^{3} T^{5} - 3581 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 162 T + 128770 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 721 T + 288540 T^{2} - 721 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 216 T - 110122 T^{2} + 10987488 T^{3} + 8956160067 T^{4} + 10987488 p^{3} T^{5} - 110122 p^{6} T^{6} - 216 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 495 T - 110077 T^{2} - 28387260 T^{3} + 67454482350 T^{4} - 28387260 p^{3} T^{5} - 110077 p^{6} T^{6} - 495 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 173 T - 252777 T^{2} - 22152996 T^{3} + 31595360704 T^{4} - 22152996 p^{3} T^{5} - 252777 p^{6} T^{6} + 173 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 532 T - 164494 T^{2} - 3428208 T^{3} + 84510915419 T^{4} - 3428208 p^{3} T^{5} - 164494 p^{6} T^{6} + 532 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 111 T - 306211 T^{2} - 31412334 T^{3} + 7298551932 T^{4} - 31412334 p^{3} T^{5} - 306211 p^{6} T^{6} + 111 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1215 T + 464669 T^{2} - 283729230 T^{3} + 297634694022 T^{4} - 283729230 p^{3} T^{5} + 464669 p^{6} T^{6} - 1215 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1460 T + 798095 T^{2} + 507243420 T^{3} + 484186197104 T^{4} + 507243420 p^{3} T^{5} + 798095 p^{6} T^{6} + 1460 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1409 T + 1147696 T^{2} + 1409 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1974 T + 1597682 T^{2} - 1754996544 T^{3} + 2041356338031 T^{4} - 1754996544 p^{3} T^{5} + 1597682 p^{6} T^{6} - 1974 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 561 T + 1863448 T^{2} + 561 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 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
−6.06635289375375711430403467263, −6.02117657769827667713715245093, −5.95065663107156482050883135034, −5.80475584750062835761035579293, −5.14250588282257076259255072644, −5.11645454525973998436491332834, −5.09018147921569842974163051985, −4.71770880556771709888292946716, −4.31267797096039500527496983914, −4.22787554441506445415742962051, −4.18567407459799671968933352342, −3.78257080788089490356439300538, −3.76251976209468732228314799642, −3.33795765359656333893010673073, −2.88309018104490987649317037707, −2.75864214222609468147145653552, −2.62938410518615903235292227433, −2.54670224662013117933191134006, −2.10357420487986552593820261869, −1.48344129938088278693947755420, −1.47151720404592913041475076776, −1.17828737200704121073959289022, −0.847854176775164903666087219162, −0.31492021047293474668080309868, −0.22630158556923908859196100968,
0.22630158556923908859196100968, 0.31492021047293474668080309868, 0.847854176775164903666087219162, 1.17828737200704121073959289022, 1.47151720404592913041475076776, 1.48344129938088278693947755420, 2.10357420487986552593820261869, 2.54670224662013117933191134006, 2.62938410518615903235292227433, 2.75864214222609468147145653552, 2.88309018104490987649317037707, 3.33795765359656333893010673073, 3.76251976209468732228314799642, 3.78257080788089490356439300538, 4.18567407459799671968933352342, 4.22787554441506445415742962051, 4.31267797096039500527496983914, 4.71770880556771709888292946716, 5.09018147921569842974163051985, 5.11645454525973998436491332834, 5.14250588282257076259255072644, 5.80475584750062835761035579293, 5.95065663107156482050883135034, 6.02117657769827667713715245093, 6.06635289375375711430403467263
