| L(s) = 1 | + 8·11-s − 16·13-s − 8·37-s − 48·47-s + 32·59-s − 40·61-s − 24·83-s + 64·97-s − 16·107-s − 32·109-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 128·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 4.43·13-s − 1.31·37-s − 7.00·47-s + 4.16·59-s − 5.12·61-s − 2.63·83-s + 6.49·97-s − 1.54·107-s − 3.06·109-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02009936289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02009936289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 36 T^{4} + 1126 T^{8} - 36 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T + 8 T^{2} + 4 T^{3} - 142 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 56 T^{3} + 62 T^{4} + 56 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + 274 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 612 T^{4} - 384602 T^{8} - 612 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 80 T^{2} + 3074 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 120 T^{2} + 6514 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 53 | \( 1 - 4388 T^{4} + 10107366 T^{8} - 4388 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 16 T + 128 T^{2} - 560 T^{3} + 1874 T^{4} - 560 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 20 T + 200 T^{2} + 1100 T^{3} + 6014 T^{4} + 1100 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 4516 T^{4} + 43788006 T^{8} + 4516 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 100 T^{2} + 5414 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 60 T^{2} + 9766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 48 T^{2} + 9026 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 12 T + 72 T^{2} - 300 T^{3} - 10414 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 144 T^{2} + 9826 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 230 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.67987286660209338930628076660, −3.62990224156568803687136199173, −3.52458269589829744157146973782, −3.46469912532741221688463464568, −3.27169502290219397328241772892, −3.05434217653632499289582829032, −2.91340981098919088820032023821, −2.91302280461639818312165637377, −2.89412801163881364953531421641, −2.78406933647825367410263572953, −2.48750815081801553129709224391, −2.32878331314280958794470295795, −2.20566796433822571924417780054, −2.15527810656692774698075429971, −1.88823607206170549291145534669, −1.87723647096556235121743694403, −1.65913639262212216592095897357, −1.39706366232361163389130995758, −1.38117923257569081981834958958, −1.35482153347143538614748886056, −1.29050067723951266510153439638, −0.61018952234796404060463052014, −0.45523259228821677282474347905, −0.43812598457307346397781905023, −0.01710634223782635842257189664,
0.01710634223782635842257189664, 0.43812598457307346397781905023, 0.45523259228821677282474347905, 0.61018952234796404060463052014, 1.29050067723951266510153439638, 1.35482153347143538614748886056, 1.38117923257569081981834958958, 1.39706366232361163389130995758, 1.65913639262212216592095897357, 1.87723647096556235121743694403, 1.88823607206170549291145534669, 2.15527810656692774698075429971, 2.20566796433822571924417780054, 2.32878331314280958794470295795, 2.48750815081801553129709224391, 2.78406933647825367410263572953, 2.89412801163881364953531421641, 2.91302280461639818312165637377, 2.91340981098919088820032023821, 3.05434217653632499289582829032, 3.27169502290219397328241772892, 3.46469912532741221688463464568, 3.52458269589829744157146973782, 3.62990224156568803687136199173, 3.67987286660209338930628076660
Plot not available for L-functions of degree greater than 10.
