| L(s) = 1 | − 16·4-s + 24·9-s + 96·11-s + 144·16-s + 12·25-s + 64·29-s − 384·36-s − 1.53e3·44-s + 112·49-s − 960·64-s − 384·71-s − 608·79-s + 324·81-s + 2.30e3·99-s − 192·100-s − 112·109-s − 1.02e3·116-s + 3.71e3·121-s + 127-s + 131-s + 137-s + 139-s + 3.45e3·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4·4-s + 8/3·9-s + 8.72·11-s + 9·16-s + 0.479·25-s + 2.20·29-s − 10.6·36-s − 34.9·44-s + 16/7·49-s − 15·64-s − 5.40·71-s − 7.69·79-s + 4·81-s + 23.2·99-s − 1.91·100-s − 1.02·109-s − 8.82·116-s + 30.6·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 24·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9609536134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9609536134\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{8} \) |
| 3 | \( ( 1 - p T^{2} )^{8} \) |
| 5 | \( 1 - 12 T^{2} - 376 T^{4} - 132 p^{2} T^{6} + 942 p^{4} T^{8} - 132 p^{6} T^{10} - 376 p^{8} T^{12} - 12 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( 1 - 16 p T^{2} + 3932 T^{4} - 24720 T^{6} - 24154 p^{2} T^{8} - 24720 p^{4} T^{10} + 3932 p^{8} T^{12} - 16 p^{13} T^{14} + p^{16} T^{16} \) |
| good | 11 | \( ( 1 - 24 T + 512 T^{2} - 6936 T^{3} + 89778 T^{4} - 6936 p^{2} T^{5} + 512 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 + 308 T^{2} + 54824 T^{4} + 3243420 T^{6} - 106725490 T^{8} + 3243420 p^{4} T^{10} + 54824 p^{8} T^{12} + 308 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 1316 T^{2} + 51880 p T^{4} + 398466540 T^{6} + 132906424526 T^{8} + 398466540 p^{4} T^{10} + 51880 p^{9} T^{12} + 1316 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 1092 T^{2} + 576680 T^{4} - 222586380 T^{6} + 81331407246 T^{8} - 222586380 p^{4} T^{10} + 576680 p^{8} T^{12} - 1092 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1300 T^{2} + 1274456 T^{4} - 947494620 T^{6} + 528617435630 T^{8} - 947494620 p^{4} T^{10} + 1274456 p^{8} T^{12} - 1300 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 16 T + 44 p T^{2} - 55408 T^{3} + 751270 T^{4} - 55408 p^{2} T^{5} + 44 p^{5} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 3948 T^{2} + 5903432 T^{4} - 3942646212 T^{6} + 1976573418510 T^{8} - 3942646212 p^{4} T^{10} + 5903432 p^{8} T^{12} - 3948 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 - 2416 T^{2} + 5835932 T^{4} - 9959371920 T^{6} + 14384225705606 T^{8} - 9959371920 p^{4} T^{10} + 5835932 p^{8} T^{12} - 2416 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 12516 T^{2} + 70001768 T^{4} - 228356671020 T^{6} + 475054496880078 T^{8} - 228356671020 p^{4} T^{10} + 70001768 p^{8} T^{12} - 12516 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 43 | \( ( 1 - 1912 T^{2} + 2635964 T^{4} - 3280241736 T^{6} + 4054697588678 T^{8} - 3280241736 p^{4} T^{10} + 2635964 p^{8} T^{12} - 1912 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 144 p T^{2} + 28283420 T^{4} + 81813152400 T^{6} + 198252999971526 T^{8} + 81813152400 p^{4} T^{10} + 28283420 p^{8} T^{12} + 144 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 6868 T^{2} + 39779480 T^{4} - 162177493980 T^{6} + 178595339294 p^{2} T^{8} - 162177493980 p^{4} T^{10} + 39779480 p^{8} T^{12} - 6868 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 11736 T^{2} + 79787516 T^{4} - 412369170408 T^{6} + 1633889740528710 T^{8} - 412369170408 p^{4} T^{10} + 79787516 p^{8} T^{12} - 11736 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 15944 T^{2} + 96425692 T^{4} - 269693317496 T^{6} + 596407807846918 T^{8} - 269693317496 p^{4} T^{10} + 96425692 p^{8} T^{12} - 15944 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 - 18688 T^{2} + 141351740 T^{4} - 579001624320 T^{6} + 2070725817126086 T^{8} - 579001624320 p^{4} T^{10} + 141351740 p^{8} T^{12} - 18688 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 + 96 T + 14048 T^{2} + 646080 T^{3} + 72093138 T^{4} + 646080 p^{2} T^{5} + 14048 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 + 18612 T^{2} + 177429992 T^{4} + 1152928560348 T^{6} + 6336699932034510 T^{8} + 1152928560348 p^{4} T^{10} + 177429992 p^{8} T^{12} + 18612 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 + 152 T + 28916 T^{2} + 2541000 T^{3} + 270417254 T^{4} + 2541000 p^{2} T^{5} + 28916 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 39632 T^{2} + 756012572 T^{4} + 9074582537520 T^{6} + 74694570513655814 T^{8} + 9074582537520 p^{4} T^{10} + 756012572 p^{8} T^{12} + 39632 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 38508 T^{2} + 778838024 T^{4} - 10318651040964 T^{6} + 96273467949842958 T^{8} - 10318651040964 p^{4} T^{10} + 778838024 p^{8} T^{12} - 38508 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 + 54804 T^{2} + 1439825768 T^{4} + 23691050976252 T^{6} + 266516143722806094 T^{8} + 23691050976252 p^{4} T^{10} + 1439825768 p^{8} T^{12} + 54804 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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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
−3.44711580464565117195048397591, −3.33524501758920191457286235803, −3.24740825352504877813851868148, −3.18952039242103800300050060939, −2.94401261777563716786596863987, −2.84085022297549423779423461073, −2.64250081782926498525227138006, −2.57112856689765423782135269991, −2.55230901136752530224008404718, −2.48797164005557841424733456838, −2.25483381429236291908954556076, −2.04299729567288167162465157691, −1.71394249144172783993118118325, −1.59792226917437829863083949285, −1.55422702158172108871095695631, −1.46099022246911266346196111961, −1.32905926768523403191957811108, −1.31905674929647061939535401690, −1.22024239900995042286166793687, −1.10443463694290013294539137599, −1.08655043194595114141011575119, −0.943754142650503365375538419095, −0.76768804751906687779987661325, −0.19324878262577795728968904115, −0.10301537076912037164693067707,
0.10301537076912037164693067707, 0.19324878262577795728968904115, 0.76768804751906687779987661325, 0.943754142650503365375538419095, 1.08655043194595114141011575119, 1.10443463694290013294539137599, 1.22024239900995042286166793687, 1.31905674929647061939535401690, 1.32905926768523403191957811108, 1.46099022246911266346196111961, 1.55422702158172108871095695631, 1.59792226917437829863083949285, 1.71394249144172783993118118325, 2.04299729567288167162465157691, 2.25483381429236291908954556076, 2.48797164005557841424733456838, 2.55230901136752530224008404718, 2.57112856689765423782135269991, 2.64250081782926498525227138006, 2.84085022297549423779423461073, 2.94401261777563716786596863987, 3.18952039242103800300050060939, 3.24740825352504877813851868148, 3.33524501758920191457286235803, 3.44711580464565117195048397591
Plot not available for L-functions of degree greater than 10.
