| L(s) = 1 | − 6·3-s − 11·5-s + 6·7-s + 9·9-s + 5·11-s + 10·13-s + 66·15-s − 100·17-s − 67·19-s − 36·21-s − 76·23-s + 232·25-s + 54·27-s + 550·29-s − 362·31-s − 30·33-s − 66·35-s + 5·37-s − 60·39-s − 324·41-s + 1.44e3·43-s − 99·45-s + 216·47-s + 113·49-s + 600·51-s − 495·53-s − 55·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.983·5-s + 0.323·7-s + 1/3·9-s + 0.137·11-s + 0.213·13-s + 1.13·15-s − 1.42·17-s − 0.808·19-s − 0.374·21-s − 0.689·23-s + 1.85·25-s + 0.384·27-s + 3.52·29-s − 2.09·31-s − 0.158·33-s − 0.318·35-s + 0.0222·37-s − 0.246·39-s − 1.23·41-s + 5.11·43-s − 0.327·45-s + 0.670·47-s + 0.329·49-s + 1.64·51-s − 1.28·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.185698286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185698286\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 6 T - 11 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 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 | $D_{4}$ | \( ( 1 - 561 T + 1863448 T^{2} - 561 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−10.21105947490999669918708061280, −9.853226891659297405119168916251, −9.544096545418988071152647667057, −8.816554730569773242426660490544, −8.808011714753263927219064747200, −8.780437512202873163813447330148, −8.285602343179141403045804233851, −8.007761348590916812435360310504, −7.44595059800541147959507452981, −7.15704491940414772297050927207, −7.10204268317453016836712492321, −6.41534323992155094725804974663, −6.40668805022569083674497547561, −5.82313109635435076567666520079, −5.78835256896650490276250342844, −4.91413425486300041750861112943, −4.87648557118163796343368325133, −4.48912220867212655707010542272, −3.99985911549111170670009956791, −3.85246432158420167211907892682, −2.94545255976121480248048515171, −2.59954412378196091578593228608, −1.95768556703895294904276176935, −0.895062648107257880001035876520, −0.51754181997242148269473396959,
0.51754181997242148269473396959, 0.895062648107257880001035876520, 1.95768556703895294904276176935, 2.59954412378196091578593228608, 2.94545255976121480248048515171, 3.85246432158420167211907892682, 3.99985911549111170670009956791, 4.48912220867212655707010542272, 4.87648557118163796343368325133, 4.91413425486300041750861112943, 5.78835256896650490276250342844, 5.82313109635435076567666520079, 6.40668805022569083674497547561, 6.41534323992155094725804974663, 7.10204268317453016836712492321, 7.15704491940414772297050927207, 7.44595059800541147959507452981, 8.007761348590916812435360310504, 8.285602343179141403045804233851, 8.780437512202873163813447330148, 8.808011714753263927219064747200, 8.816554730569773242426660490544, 9.544096545418988071152647667057, 9.853226891659297405119168916251, 10.21105947490999669918708061280
