| L(s) = 1 | − 20·9-s − 16·19-s − 240·31-s + 328·49-s + 16·61-s + 176·79-s + 312·81-s + 432·109-s + 160·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.15e3·169-s + 320·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.22·9-s − 0.842·19-s − 7.74·31-s + 6.69·49-s + 0.262·61-s + 2.22·79-s + 3.85·81-s + 3.96·109-s + 1.32·121-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 + 0.00598·167-s + 6.81·169-s + 1.87·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1712910195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1712910195\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 20 T^{2} + 88 T^{4} - 460 p T^{6} - 2498 p^{2} T^{8} - 460 p^{5} T^{10} + 88 p^{8} T^{12} + 20 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 164 T^{2} + 16984 T^{4} - 1225804 T^{6} + 69410030 T^{8} - 1225804 p^{4} T^{10} + 16984 p^{8} T^{12} - 164 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 - 80 T^{2} - 8612 T^{4} + 1315920 T^{6} + 21445638 T^{8} + 1315920 p^{4} T^{10} - 8612 p^{8} T^{12} - 80 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 - 576 T^{2} + 179516 T^{4} - 39840192 T^{6} + 7152638406 T^{8} - 39840192 p^{4} T^{10} + 179516 p^{8} T^{12} - 576 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 1208 T^{2} + 736924 T^{4} + 304675080 T^{6} + 97385661510 T^{8} + 304675080 p^{4} T^{10} + 736924 p^{8} T^{12} + 1208 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 + 4 T + 736 T^{2} + 908 T^{3} + 287486 T^{4} + 908 p^{2} T^{5} + 736 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 23 | \( ( 1 + 1268 T^{2} + 1124824 T^{4} + 670011900 T^{6} + 391776576750 T^{8} + 670011900 p^{4} T^{10} + 1124824 p^{8} T^{12} + 1268 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 4352 T^{2} + 8975164 T^{4} - 11861548800 T^{6} + 11435522898630 T^{8} - 11861548800 p^{4} T^{10} + 8975164 p^{8} T^{12} - 4352 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 + 60 T + 2920 T^{2} + 101620 T^{3} + 3613902 T^{4} + 101620 p^{2} T^{5} + 2920 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 - 4480 T^{2} + 9991228 T^{4} - 18818637440 T^{6} + 30040006766918 T^{8} - 18818637440 p^{4} T^{10} + 9991228 p^{8} T^{12} - 4480 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 1920 T^{2} + 9436924 T^{4} - 13365676160 T^{6} + 37834324951686 T^{8} - 13365676160 p^{4} T^{10} + 9436924 p^{8} T^{12} - 1920 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 43 | \( ( 1 - 6860 T^{2} + 28453528 T^{4} - 81867044740 T^{6} + 174047931980078 T^{8} - 81867044740 p^{4} T^{10} + 28453528 p^{8} T^{12} - 6860 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 15060 T^{2} + 103552984 T^{4} + 425803156380 T^{6} + 1147648197554286 T^{8} + 425803156380 p^{4} T^{10} + 103552984 p^{8} T^{12} + 15060 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 + 5496 T^{2} + 16907740 T^{4} + 66116431304 T^{6} + 230534565871494 T^{8} + 66116431304 p^{4} T^{10} + 16907740 p^{8} T^{12} + 5496 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 15232 T^{2} + 125199548 T^{4} - 692185732224 T^{6} + 2791710943429190 T^{8} - 692185732224 p^{4} T^{10} + 125199548 p^{8} T^{12} - 15232 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 4 T + 2376 T^{2} - 32108 T^{3} + 20621806 T^{4} - 32108 p^{2} T^{5} + 2376 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 - 20428 T^{2} + 205316248 T^{4} - 1379854106116 T^{6} + 6997703881378990 T^{8} - 1379854106116 p^{4} T^{10} + 205316248 p^{8} T^{12} - 20428 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 24024 T^{2} + 302202556 T^{4} - 2503345638248 T^{6} + 14778735683172486 T^{8} - 2503345638248 p^{4} T^{10} + 302202556 p^{8} T^{12} - 24024 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 35320 T^{2} + 580621468 T^{4} - 5749042183880 T^{6} + 37382001510683078 T^{8} - 5749042183880 p^{4} T^{10} + 580621468 p^{8} T^{12} - 35320 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 44 T + 23696 T^{2} - 832068 T^{3} + 217860926 T^{4} - 832068 p^{2} T^{5} + 23696 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 13396 T^{2} + 137438936 T^{4} + 1239330480732 T^{6} + 8621185666547246 T^{8} + 1239330480732 p^{4} T^{10} + 137438936 p^{8} T^{12} + 13396 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 14056 T^{2} + 62829020 T^{4} + 214903235496 T^{6} - 3856000255517626 T^{8} + 214903235496 p^{4} T^{10} + 62829020 p^{8} T^{12} - 14056 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 49336 T^{2} + 1219926940 T^{4} - 19431827300104 T^{6} + 216291237745346374 T^{8} - 19431827300104 p^{4} T^{10} + 1219926940 p^{8} T^{12} - 49336 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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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
−2.30237441879602588640467696346, −2.25158739561207721520209621964, −2.23071813254587016179546976220, −2.20085045742550634746972731375, −2.01839286112473297553573198588, −1.95925465608452610749554549451, −1.87163614779058000021447205667, −1.77867149920851320737202319909, −1.63805284239829303282791350157, −1.44360508396932744329782913348, −1.42709930119404368959080536012, −1.41431816804510256343094673549, −1.19984954689582636090295158294, −1.15175485625394893951277889061, −1.07956131912953463275451581631, −1.03497852178170086435745717313, −1.02747248006787136289273415214, −0.75966842138599012861648214335, −0.58279790206938168146175737074, −0.56465841991069325837495657513, −0.44230964854628596309923360295, −0.39152991276424670778920824537, −0.21447149708571841218663622904, −0.14277750999915893079752745281, −0.03414108059897466442791895846,
0.03414108059897466442791895846, 0.14277750999915893079752745281, 0.21447149708571841218663622904, 0.39152991276424670778920824537, 0.44230964854628596309923360295, 0.56465841991069325837495657513, 0.58279790206938168146175737074, 0.75966842138599012861648214335, 1.02747248006787136289273415214, 1.03497852178170086435745717313, 1.07956131912953463275451581631, 1.15175485625394893951277889061, 1.19984954689582636090295158294, 1.41431816804510256343094673549, 1.42709930119404368959080536012, 1.44360508396932744329782913348, 1.63805284239829303282791350157, 1.77867149920851320737202319909, 1.87163614779058000021447205667, 1.95925465608452610749554549451, 2.01839286112473297553573198588, 2.20085045742550634746972731375, 2.23071813254587016179546976220, 2.25158739561207721520209621964, 2.30237441879602588640467696346
Plot not available for L-functions of degree greater than 10.
