| L(s) = 1 | − 10·4-s + 67·16-s + 128·29-s + 32·41-s − 336·49-s − 352·61-s − 300·64-s + 160·89-s + 448·101-s − 736·109-s − 1.28e3·116-s + 1.10e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 320·164-s + 167-s + 688·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 4.18·16-s + 4.41·29-s + 0.780·41-s − 6.85·49-s − 5.77·61-s − 4.68·64-s + 1.79·89-s + 4.43·101-s − 6.75·109-s − 11.0·116-s + 9.12·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 1.95·164-s + 0.00598·167-s + 4.07·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0006817175833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0006817175833\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5 p T^{2} + 33 T^{4} - 5 p^{3} T^{6} - 37 p^{4} T^{8} - 5 p^{7} T^{10} + 33 p^{8} T^{12} + 5 p^{13} T^{14} + p^{16} T^{16} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 24 p T^{2} + 13404 T^{4} + 690840 T^{6} + 31402310 T^{8} + 690840 p^{4} T^{10} + 13404 p^{8} T^{12} + 24 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 - 344 T^{2} + 68124 T^{4} - 13074664 T^{6} + 2372591750 T^{8} - 13074664 p^{4} T^{10} + 68124 p^{8} T^{12} - 344 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 - 1464 T^{2} + 1030300 T^{4} - 469485576 T^{6} + 156060299334 T^{8} - 469485576 p^{4} T^{10} + 1030300 p^{8} T^{12} - 1464 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 + 616 T^{2} + 755676 T^{4} + 531969752 T^{6} + 267063306566 T^{8} + 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} + 616 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 32 T + 1212 T^{2} - 46048 T^{3} + 1958438 T^{4} - 46048 p^{2} T^{5} + 1212 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 - 4760 T^{2} + 9635868 T^{4} - 11236659880 T^{6} + 12108314450438 T^{8} - 11236659880 p^{4} T^{10} + 9635868 p^{8} T^{12} - 4760 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 8 T + 4924 T^{2} - 33080 T^{3} + 10990150 T^{4} - 33080 p^{2} T^{5} + 4924 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 + 3976 T^{2} + 15992796 T^{4} + 34977997112 T^{6} + 80511749221766 T^{8} + 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} + 3976 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 9640 T^{2} + 45901788 T^{4} + 144992767640 T^{6} + 353863561499078 T^{8} + 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} + 9640 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 11000 T^{2} + 64936668 T^{4} - 267744514120 T^{6} + 842750699414918 T^{8} - 267744514120 p^{4} T^{10} + 64936668 p^{8} T^{12} - 11000 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 + 88 T + 12348 T^{2} + 708776 T^{3} + 62059430 T^{4} + 708776 p^{2} T^{5} + 12348 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 + 19848 T^{2} + 189118044 T^{4} + 1228457910840 T^{6} + 6186195617725190 T^{8} + 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} + 19848 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 25080 T^{2} + 324571804 T^{4} - 2808004964040 T^{6} + 17495806969152966 T^{8} - 2808004964040 p^{4} T^{10} + 324571804 p^{8} T^{12} - 25080 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 83 | \( ( 1 + 18184 T^{2} + 266916444 T^{4} + 2437903506104 T^{6} + 19925145362261510 T^{8} + 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} + 18184 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 40 T + 11100 T^{2} - 192920 T^{3} + 121014662 T^{4} - 192920 p^{2} T^{5} + 11100 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 40376 T^{2} + 923595676 T^{4} - 13887638321672 T^{6} + 152910672794232646 T^{8} - 13887638321672 p^{4} T^{10} + 923595676 p^{8} T^{12} - 40376 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.37677219822765580275356721615, −2.32844340303710744713901730809, −2.30688149751199091522452175065, −2.25270646342606759852462488599, −2.23658791972678349119278391019, −1.95352804442867363637546423225, −1.68683072704870813401642982390, −1.61076987032136312434815271745, −1.60074855774131845202938647051, −1.55267813567289370148426460033, −1.52613509498636396686310764277, −1.51381992932725908376413306316, −1.42105954312466276784817795128, −1.27937828403310063930989669324, −1.27922306576496451974878956432, −1.06980260750143217135890745169, −0.915022416472905920345129770646, −0.69145137482740411076703516606, −0.68351523495761217914757990975, −0.61894371333456714200648948844, −0.61862525551458326742983505573, −0.46027560854055808382078930367, −0.32399462583849729220869248532, −0.03788805528801830638275087112, −0.00671605632877191305171607955,
0.00671605632877191305171607955, 0.03788805528801830638275087112, 0.32399462583849729220869248532, 0.46027560854055808382078930367, 0.61862525551458326742983505573, 0.61894371333456714200648948844, 0.68351523495761217914757990975, 0.69145137482740411076703516606, 0.915022416472905920345129770646, 1.06980260750143217135890745169, 1.27922306576496451974878956432, 1.27937828403310063930989669324, 1.42105954312466276784817795128, 1.51381992932725908376413306316, 1.52613509498636396686310764277, 1.55267813567289370148426460033, 1.60074855774131845202938647051, 1.61076987032136312434815271745, 1.68683072704870813401642982390, 1.95352804442867363637546423225, 2.23658791972678349119278391019, 2.25270646342606759852462488599, 2.30688149751199091522452175065, 2.32844340303710744713901730809, 2.37677219822765580275356721615
Plot not available for L-functions of degree greater than 10.
