| L(s) = 1 | − 8·5-s − 24·13-s + 8·17-s + 4·25-s − 136·29-s + 40·37-s − 8·41-s + 100·49-s + 88·53-s + 296·61-s + 192·65-s + 88·73-s − 64·85-s − 216·89-s − 328·97-s + 24·101-s − 440·109-s + 312·113-s + 140·121-s + 72·125-s + 127-s + 131-s + 137-s + 139-s + 1.08e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 1.84·13-s + 8/17·17-s + 4/25·25-s − 4.68·29-s + 1.08·37-s − 0.195·41-s + 2.04·49-s + 1.66·53-s + 4.85·61-s + 2.95·65-s + 1.20·73-s − 0.752·85-s − 2.42·89-s − 3.38·97-s + 0.237·101-s − 4.03·109-s + 2.76·113-s + 1.15·121-s + 0.575·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.50·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8907034353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8907034353\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5254 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 5382 T^{4} - 140 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 12 T + 342 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 454 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 780 T^{2} + 402374 T^{4} - 780 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 484 T^{2} + 517894 T^{4} - 484 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 68 T + 2806 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 20 T + 1270 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 2854 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2764 T^{2} + 8710534 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7428 T^{2} + 23258246 T^{4} - 7428 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 44 T + 6070 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11020 T^{2} + 52720774 T^{4} - 11020 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 148 T + 11350 T^{2} - 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 11980 T^{2} + 75342534 T^{4} - 11980 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18276 T^{2} + 133865606 T^{4} - 18276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 44 T + 9990 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1028 T^{2} + 28324230 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5452 T^{2} + 97966918 T^{4} - 5452 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 108 T + 15558 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 164 T + 22342 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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
−6.79237435667556819056643929215, −6.71075810099601137739577856587, −6.67569389270768215439594688134, −5.92778155714145176842298697262, −5.79390213522096221376408592971, −5.49233862532290499871106146429, −5.49182804523291770368751606873, −5.42646902145214679156303031616, −5.11645030300561293873492710694, −4.58308482587066920106240595792, −4.48947330184999427874938858133, −4.18136666892342915375042209231, −3.81359042992171121528247121536, −3.80158161839268499609474282273, −3.70344118577338529385707351447, −3.52941693951995641179996348667, −2.75492701746988251290542984514, −2.73210282463609309240009677697, −2.32172169616497785474686100665, −2.23725405442507312031015686446, −1.85961928747459417730076731839, −1.37954111021008924505252364283, −0.953225677013276026226056673200, −0.40523535006971082673779612066, −0.25851254791083164114166383014,
0.25851254791083164114166383014, 0.40523535006971082673779612066, 0.953225677013276026226056673200, 1.37954111021008924505252364283, 1.85961928747459417730076731839, 2.23725405442507312031015686446, 2.32172169616497785474686100665, 2.73210282463609309240009677697, 2.75492701746988251290542984514, 3.52941693951995641179996348667, 3.70344118577338529385707351447, 3.80158161839268499609474282273, 3.81359042992171121528247121536, 4.18136666892342915375042209231, 4.48947330184999427874938858133, 4.58308482587066920106240595792, 5.11645030300561293873492710694, 5.42646902145214679156303031616, 5.49182804523291770368751606873, 5.49233862532290499871106146429, 5.79390213522096221376408592971, 5.92778155714145176842298697262, 6.67569389270768215439594688134, 6.71075810099601137739577856587, 6.79237435667556819056643929215
