| L(s) = 1 | + 12·2-s + 72·4-s + 72·7-s + 264·8-s + 24·11-s − 144·13-s + 864·14-s + 220·16-s + 600·17-s + 288·22-s + 888·23-s − 1.72e3·26-s + 5.18e3·28-s + 3.24e3·31-s − 4.32e3·32-s + 7.20e3·34-s − 2.44e3·37-s − 6.86e3·41-s + 5.54e3·43-s + 1.72e3·44-s + 1.06e4·46-s + 5.61e3·47-s + 2.59e3·49-s − 1.03e4·52-s + 1.84e3·53-s + 1.90e4·56-s + 1.50e4·61-s + ⋯ |
| L(s) = 1 | + 3·2-s + 9/2·4-s + 1.46·7-s + 33/8·8-s + 0.198·11-s − 0.852·13-s + 4.40·14-s + 0.859·16-s + 2.07·17-s + 0.595·22-s + 1.67·23-s − 2.55·26-s + 6.61·28-s + 3.37·31-s − 4.21·32-s + 6.22·34-s − 1.78·37-s − 4.08·41-s + 2.99·43-s + 0.892·44-s + 5.03·46-s + 2.54·47-s + 1.07·49-s − 3.83·52-s + 0.657·53-s + 6.06·56-s + 4.03·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(50.74038706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(50.74038706\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 p^{2} T + 9 p^{3} T^{2} - 33 p^{3} T^{3} + 233 p^{2} T^{4} - 33 p^{7} T^{5} + 9 p^{11} T^{6} - 3 p^{14} T^{7} + p^{16} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 72 T + 2592 T^{2} - 219312 T^{3} + 18140207 T^{4} - 219312 p^{4} T^{5} + 2592 p^{8} T^{6} - 72 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 12 T - 7186 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 2429280 T^{3} + 432514319 T^{4} + 2429280 p^{4} T^{5} + 10368 p^{8} T^{6} + 144 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 600 T + 180000 T^{2} - 77105400 T^{3} + 31005206018 T^{4} - 77105400 p^{4} T^{5} + 180000 p^{8} T^{6} - 600 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 326690 T^{2} + 54622417923 T^{4} - 326690 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 888 T + 394272 T^{2} - 52907928 T^{3} - 41414676478 T^{4} - 52907928 p^{4} T^{5} + 394272 p^{8} T^{6} - 888 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 63932 p T^{2} + 1839990957414 T^{4} - 63932 p^{9} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 1622 T + 2141667 T^{2} - 1622 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2448 T + 2996352 T^{2} + 6308792208 T^{3} + 12788952535682 T^{4} + 6308792208 p^{4} T^{5} + 2996352 p^{8} T^{6} + 2448 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3432 T + 8077562 T^{2} + 3432 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5544 T + 15367968 T^{2} - 34365692592 T^{3} + 69120271160159 T^{4} - 34365692592 p^{4} T^{5} + 15367968 p^{8} T^{6} - 5544 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5616 T + 15769728 T^{2} - 24055647408 T^{3} + 36339718329314 T^{4} - 24055647408 p^{4} T^{5} + 15769728 p^{8} T^{6} - 5616 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1848 T + 1707552 T^{2} - 12837022968 T^{3} + 95614873611362 T^{4} - 12837022968 p^{4} T^{5} + 1707552 p^{8} T^{6} - 1848 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10941820 T^{2} + 100216939084998 T^{4} - 10941820 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7510 T + 35401563 T^{2} - 7510 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 13320 T + 88711200 T^{2} - 439054559280 T^{3} + 2008874003467343 T^{4} - 439054559280 p^{4} T^{5} + 88711200 p^{8} T^{6} - 13320 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10776 T + 74664506 T^{2} - 10776 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 19728 T + 194596992 T^{2} + 1434198789648 T^{3} + 8607659366333762 T^{4} + 1434198789648 p^{4} T^{5} + 194596992 p^{8} T^{6} + 19728 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 25715324 T^{2} + 2139409705359366 T^{4} - 25715324 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8592 T + 36911232 T^{2} - 464368361424 T^{3} + 5798663913908642 T^{4} - 464368361424 p^{4} T^{5} + 36911232 p^{8} T^{6} - 8592 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 203730340 T^{2} + 18051837318885318 T^{4} - 203730340 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11520 T + 66355200 T^{2} - 1104271879680 T^{3} + 18323409093272303 T^{4} - 1104271879680 p^{4} T^{5} + 66355200 p^{8} T^{6} - 11520 p^{12} T^{7} + p^{16} T^{8} \) |
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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
−8.054311115242487607532050766108, −7.996739996110687215885716607190, −7.64654957873019100362754269719, −7.07955080256335008454405457484, −6.94205133341413268449924494944, −6.91192029295363021046411294321, −6.48984348406326591501275746595, −6.37489259587967193353546871011, −5.67529987251704330544222928728, −5.32868869570267731226746474207, −5.26073288733230809350680659048, −5.19777550159643541388472578213, −5.13057746396331239208092653071, −4.54623242134844720468218902561, −4.15146561513585772842127386224, −4.05102625381307639501136609207, −3.86712874276154063902563565289, −3.22699057152293217961236324659, −3.09548567616509911095838935387, −2.62771321194785946731997675989, −2.20223545361543766403679844943, −2.16968833048115752995714645196, −1.32508846540233992283577987006, −0.856688728607763119562881801805, −0.65627087011487144786381102320,
0.65627087011487144786381102320, 0.856688728607763119562881801805, 1.32508846540233992283577987006, 2.16968833048115752995714645196, 2.20223545361543766403679844943, 2.62771321194785946731997675989, 3.09548567616509911095838935387, 3.22699057152293217961236324659, 3.86712874276154063902563565289, 4.05102625381307639501136609207, 4.15146561513585772842127386224, 4.54623242134844720468218902561, 5.13057746396331239208092653071, 5.19777550159643541388472578213, 5.26073288733230809350680659048, 5.32868869570267731226746474207, 5.67529987251704330544222928728, 6.37489259587967193353546871011, 6.48984348406326591501275746595, 6.91192029295363021046411294321, 6.94205133341413268449924494944, 7.07955080256335008454405457484, 7.64654957873019100362754269719, 7.996739996110687215885716607190, 8.054311115242487607532050766108
