| L(s) = 1 | − 2·4-s − 12·7-s + 4·13-s + 16·16-s + 44·25-s + 24·28-s + 96·31-s − 56·37-s + 240·43-s + 2·49-s − 8·52-s − 20·61-s − 88·64-s − 228·67-s + 232·73-s − 660·79-s − 48·91-s + 124·97-s − 88·100-s + 348·103-s − 560·109-s − 192·112-s − 124·121-s − 192·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.71·7-s + 4/13·13-s + 16-s + 1.75·25-s + 6/7·28-s + 3.09·31-s − 1.51·37-s + 5.58·43-s + 2/49·49-s − 0.153·52-s − 0.327·61-s − 1.37·64-s − 3.40·67-s + 3.17·73-s − 8.35·79-s − 0.527·91-s + 1.27·97-s − 0.879·100-s + 3.37·103-s − 5.13·109-s − 1.71·112-s − 1.02·121-s − 1.54·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1376473064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1376473064\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T^{2} - 3 p^{2} T^{4} + p^{5} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 44 T^{2} + 922 T^{4} + 10384 T^{6} - 572429 T^{8} + 10384 p^{4} T^{10} + 922 p^{8} T^{12} - 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 6 T + 53 T^{2} + 246 T^{3} - 132 T^{4} + 246 p^{2} T^{5} + 53 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 124 T^{2} + 250 T^{4} - 1755344 T^{6} - 210683021 T^{8} - 1755344 p^{4} T^{10} + 250 p^{8} T^{12} + 124 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 2 T - 155 T^{2} + 358 T^{3} - 3956 T^{4} + 358 p^{2} T^{5} - 155 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 140 T^{2} + 136662 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 695 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 604 T^{2} - 268070 T^{4} + 44215216 T^{6} + 220143229459 T^{8} + 44215216 p^{4} T^{10} - 268070 p^{8} T^{12} + 604 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 2468 T^{2} + 3338026 T^{4} - 3303260048 T^{6} + 2770686029299 T^{8} - 3303260048 p^{4} T^{10} + 3338026 p^{8} T^{12} - 2468 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 48 T + 2 p^{2} T^{2} - 55392 T^{3} + 1146243 T^{4} - 55392 p^{2} T^{5} + 2 p^{6} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 14 T + 2607 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 - 3140 T^{2} + 4692298 T^{4} + 1520450800 T^{6} - 9144788425517 T^{8} + 1520450800 p^{4} T^{10} + 4692298 p^{8} T^{12} - 3140 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 120 T + 9458 T^{2} - 558960 T^{3} + 27153363 T^{4} - 558960 p^{2} T^{5} + 9458 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 4444 T^{2} + 113030 p T^{4} + 20786205616 T^{6} + 88238287231699 T^{8} + 20786205616 p^{4} T^{10} + 113030 p^{9} T^{12} + 4444 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 508 T^{2} + 14912358 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 6076 T^{2} + 12975610 T^{4} - 1777570256 T^{6} - 20949143941901 T^{8} - 1777570256 p^{4} T^{10} + 12975610 p^{8} T^{12} + 6076 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 10 T - 7187 T^{2} - 1550 T^{3} + 38882428 T^{4} - 1550 p^{2} T^{5} - 7187 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 114 T + 10553 T^{2} + 709194 T^{3} + 37996068 T^{4} + 709194 p^{2} T^{5} + 10553 p^{4} T^{6} + 114 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 292 T^{2} + 42446598 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 58 T + 10779 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 330 T + 54917 T^{2} + 6143610 T^{3} + 534190908 T^{4} + 6143610 p^{2} T^{5} + 54917 p^{4} T^{6} + 330 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 26116 T^{2} + 416905450 T^{4} + 4445553374224 T^{6} + 35678198362101619 T^{8} + 4445553374224 p^{4} T^{10} + 416905450 p^{8} T^{12} + 26116 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 30668 T^{2} + 360580758 T^{4} + 30668 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 62 T - 4415 T^{2} + 654658 T^{3} - 56486396 T^{4} + 654658 p^{2} T^{5} - 4415 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−5.01907998916287142716307545246, −4.61435697269933320709599624293, −4.57832482960256113093783431757, −4.49793333023929501660879446708, −4.43540617126064433534943300960, −4.21375140630099481587966613009, −3.95235774545545307016120264434, −3.91217863337799101736891298349, −3.60308213802713160886740464618, −3.49610183042715645092110285688, −3.39825728997135922002209290237, −3.29403652917845570625225828745, −2.89681190219868419157122629770, −2.69998226695508397121394301836, −2.65457154949337291670395594915, −2.57968620643799072133218407307, −2.54130157857475170490874410089, −2.28780765618496013841311827566, −1.69372693812505844151052558216, −1.25072959117205332606473485477, −1.24723021255041196314414304748, −1.17627217650322922865146879573, −1.08900733925357888618651767168, −0.37531759173962884250762933646, −0.06192059298469976378578384720,
0.06192059298469976378578384720, 0.37531759173962884250762933646, 1.08900733925357888618651767168, 1.17627217650322922865146879573, 1.24723021255041196314414304748, 1.25072959117205332606473485477, 1.69372693812505844151052558216, 2.28780765618496013841311827566, 2.54130157857475170490874410089, 2.57968620643799072133218407307, 2.65457154949337291670395594915, 2.69998226695508397121394301836, 2.89681190219868419157122629770, 3.29403652917845570625225828745, 3.39825728997135922002209290237, 3.49610183042715645092110285688, 3.60308213802713160886740464618, 3.91217863337799101736891298349, 3.95235774545545307016120264434, 4.21375140630099481587966613009, 4.43540617126064433534943300960, 4.49793333023929501660879446708, 4.57832482960256113093783431757, 4.61435697269933320709599624293, 5.01907998916287142716307545246
Plot not available for L-functions of degree greater than 10.
