| L(s) = 1 | − 6·5-s + 10·13-s − 60·17-s + 61·25-s − 66·29-s − 40·37-s + 144·41-s − 97·49-s + 360·53-s + 14·61-s − 60·65-s − 220·73-s + 360·85-s − 912·89-s + 200·97-s − 198·101-s − 248·109-s + 570·113-s − 214·121-s − 270·125-s + 127-s + 131-s + 137-s + 139-s + 396·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 6/5·5-s + 0.769·13-s − 3.52·17-s + 2.43·25-s − 2.27·29-s − 1.08·37-s + 3.51·41-s − 1.97·49-s + 6.79·53-s + 0.229·61-s − 0.923·65-s − 3.01·73-s + 4.23·85-s − 10.2·89-s + 2.06·97-s − 1.96·101-s − 2.27·109-s + 5.04·113-s − 1.76·121-s − 2.15·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.73·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1559006373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1559006373\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 3 T - 17 T^{2} - 72 T^{3} - 174 T^{4} - 72 p^{2} T^{5} - 17 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 + 97 T^{2} + 4381 T^{4} + 21922 T^{6} - 3075026 T^{8} + 21922 p^{4} T^{10} + 4381 p^{8} T^{12} + 97 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 107 T^{2} - 3192 T^{4} + 107 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5 T - 83 T^{2} + 1150 T^{3} - 21122 T^{4} + 1150 p^{2} T^{5} - 83 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 15 T + 608 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 31 p T^{2} + 294216 T^{4} - 31 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 503 T^{2} - 197699 T^{4} + 103618 p^{2} T^{6} + 193054 p^{4} T^{8} + 103618 p^{6} T^{10} - 197699 p^{8} T^{12} - 503 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 33 T - 839 T^{2} + 8118 T^{3} + 2094054 T^{4} + 8118 p^{2} T^{5} - 839 p^{4} T^{6} + 33 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 251 T^{2} + 804865 T^{4} + 649815406 T^{6} - 360711589466 T^{8} + 649815406 p^{4} T^{10} + 804865 p^{8} T^{12} - 251 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 10 T + 1818 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 72 T + 1471 T^{2} - 25272 T^{3} + 2246304 T^{4} - 25272 p^{2} T^{5} + 1471 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 + 3166 T^{2} + 4082065 T^{4} - 2837087426 T^{6} - 10655428250876 T^{8} - 2837087426 p^{4} T^{10} + 4082065 p^{8} T^{12} + 3166 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 + 2977 T^{2} + 5326621 T^{4} - 18527222558 T^{6} - 54554955650546 T^{8} - 18527222558 p^{4} T^{10} + 5326621 p^{8} T^{12} + 2977 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 90 T + 6698 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 3587 T^{2} + 749208 T^{4} + 3587 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7 T - 7169 T^{2} + 1568 T^{3} + 38071354 T^{4} + 1568 p^{2} T^{5} - 7169 p^{4} T^{6} - 7 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8411 T^{2} + 50593800 T^{4} + 8411 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 7922 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 55 T + 5508 T^{2} + 55 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( 1 + 20869 T^{2} + 251340865 T^{4} + 2217834902446 T^{6} + 15439711699511014 T^{8} + 2217834902446 p^{4} T^{10} + 251340865 p^{8} T^{12} + 20869 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 + 16837 T^{2} + 138536161 T^{4} + 842384844142 T^{6} + 5304047075446774 T^{8} + 842384844142 p^{4} T^{10} + 138536161 p^{8} T^{12} + 16837 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 228 T + 27158 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 100 T - 29 p T^{2} + 600500 T^{3} - 18446312 T^{4} + 600500 p^{2} T^{5} - 29 p^{5} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.78046891151807239311969656792, −3.73322301472102444974700881671, −3.64639984542430179926991645596, −3.31305889775713543430505037574, −3.29942540513124194009142602075, −2.90915787475936807755259834405, −2.85534269939927656665684892853, −2.84973839994441759867209721228, −2.71081495618094996341816808665, −2.60173087509471537641655157478, −2.47268499282903299403602960536, −2.32922093327459645348042896177, −2.25201943289160056426061161677, −2.00841783542793063220924733892, −1.68712197968272805708447168116, −1.67946277772801176477727514334, −1.61209278386186234806112718698, −1.44277960640936814286221239906, −1.31633506480747887009863811431, −0.850505606236439137957866764355, −0.75049583103660921942832770521, −0.69506244724675054245724537603, −0.62223691732670770157375763182, −0.12543064962068943529062557705, −0.07065981337002726920558171651,
0.07065981337002726920558171651, 0.12543064962068943529062557705, 0.62223691732670770157375763182, 0.69506244724675054245724537603, 0.75049583103660921942832770521, 0.850505606236439137957866764355, 1.31633506480747887009863811431, 1.44277960640936814286221239906, 1.61209278386186234806112718698, 1.67946277772801176477727514334, 1.68712197968272805708447168116, 2.00841783542793063220924733892, 2.25201943289160056426061161677, 2.32922093327459645348042896177, 2.47268499282903299403602960536, 2.60173087509471537641655157478, 2.71081495618094996341816808665, 2.84973839994441759867209721228, 2.85534269939927656665684892853, 2.90915787475936807755259834405, 3.29942540513124194009142602075, 3.31305889775713543430505037574, 3.64639984542430179926991645596, 3.73322301472102444974700881671, 3.78046891151807239311969656792
Plot not available for L-functions of degree greater than 10.
