| L(s) = 1 | + 2·2-s − 2·4-s − 8·13-s + 24·16-s − 16·26-s + 32·29-s + 40·32-s + 176·37-s + 16·41-s + 204·49-s + 16·52-s − 304·53-s + 64·58-s + 136·61-s + 8·64-s − 240·73-s + 352·74-s + 32·82-s − 128·89-s − 216·97-s + 408·98-s − 112·101-s − 608·106-s − 520·109-s + 96·113-s − 64·116-s + 264·121-s + ⋯ |
| L(s) = 1 | + 2-s − 1/2·4-s − 0.615·13-s + 3/2·16-s − 0.615·26-s + 1.10·29-s + 5/4·32-s + 4.75·37-s + 0.390·41-s + 4.16·49-s + 4/13·52-s − 5.73·53-s + 1.10·58-s + 2.22·61-s + 1/8·64-s − 3.28·73-s + 4.75·74-s + 0.390·82-s − 1.43·89-s − 2.22·97-s + 4.16·98-s − 1.10·101-s − 5.73·106-s − 4.77·109-s + 0.849·113-s − 0.551·116-s + 2.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(17.20918657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.20918657\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T + 3 p T^{2} - p^{4} T^{3} + 5 p^{2} T^{4} - p^{6} T^{5} + 3 p^{5} T^{6} - p^{7} T^{7} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 204 T^{2} + 18426 T^{4} - 1076112 T^{6} + 53449475 T^{8} - 1076112 p^{4} T^{10} + 18426 p^{8} T^{12} - 204 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 24 p T^{2} + 33852 T^{4} - 3268920 T^{6} + 337315526 T^{8} - 3268920 p^{4} T^{10} + 33852 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 4 T + 258 T^{2} - 1888 T^{3} + 28523 T^{4} - 1888 p^{2} T^{5} + 258 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 324 T^{2} + 1920 T^{3} + 152630 T^{4} + 1920 p^{2} T^{5} + 324 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1756 T^{2} + 1510026 T^{4} - 44288912 p T^{6} + 346336758035 T^{8} - 44288912 p^{5} T^{10} + 1510026 p^{8} T^{12} - 1756 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 2248 T^{2} + 2472828 T^{4} - 1852675064 T^{6} + 1086691824134 T^{8} - 1852675064 p^{4} T^{10} + 2472828 p^{8} T^{12} - 2248 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 16 T + 1764 T^{2} - 10544 T^{3} + 1646102 T^{4} - 10544 p^{2} T^{5} + 1764 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 2028 T^{2} + 2805402 T^{4} - 2884712400 T^{6} + 3076391251619 T^{8} - 2884712400 p^{4} T^{10} + 2805402 p^{8} T^{12} - 2028 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 88 T + 4764 T^{2} - 139688 T^{3} + 4747046 T^{4} - 139688 p^{2} T^{5} + 4764 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 8 T + 1756 T^{2} - 78008 T^{3} + 3756598 T^{4} - 78008 p^{2} T^{5} + 1756 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 7900 T^{2} + 33132810 T^{4} - 96255124016 T^{6} + 206878197918227 T^{8} - 96255124016 p^{4} T^{10} + 33132810 p^{8} T^{12} - 7900 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 5656 T^{2} + 20883420 T^{4} - 53920616360 T^{6} + 122334534618566 T^{8} - 53920616360 p^{4} T^{10} + 20883420 p^{8} T^{12} - 5656 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 152 T + 15804 T^{2} + 1145512 T^{3} + 68059286 T^{4} + 1145512 p^{2} T^{5} + 15804 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 21656 T^{2} + 220430236 T^{4} - 1377870886568 T^{6} + 5784129582348550 T^{8} - 1377870886568 p^{4} T^{10} + 220430236 p^{8} T^{12} - 21656 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 68 T + 6786 T^{2} - 434848 T^{3} + 32554859 T^{4} - 434848 p^{2} T^{5} + 6786 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 14364 T^{2} + 117868746 T^{4} - 700523495472 T^{6} + 3348544470504275 T^{8} - 700523495472 p^{4} T^{10} + 117868746 p^{8} T^{12} - 14364 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 31512 T^{2} + 469696188 T^{4} - 4285342561320 T^{6} + 26126405445829766 T^{8} - 4285342561320 p^{4} T^{10} + 469696188 p^{8} T^{12} - 31512 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 120 T + 12732 T^{2} + 423240 T^{3} + 35299142 T^{4} + 423240 p^{2} T^{5} + 12732 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 35400 T^{2} + 591951516 T^{4} - 6232687244280 T^{6} + 45861936410406470 T^{8} - 6232687244280 p^{4} T^{10} + 591951516 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 6424 T^{2} + 64588572 T^{4} - 770632625960 T^{6} + 4303554072892550 T^{8} - 770632625960 p^{4} T^{10} + 64588572 p^{8} T^{12} - 6424 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 64 T + 28356 T^{2} + 1283264 T^{3} + 322223942 T^{4} + 1283264 p^{2} T^{5} + 28356 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 108 T + 26650 T^{2} + 2076624 T^{3} + 340059171 T^{4} + 2076624 p^{2} T^{5} + 26650 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.16661585298858824020409779673, −4.11386087778541214438847963591, −3.96071926360550514163616934721, −3.84416895268994950212143745180, −3.73564502301876154258415991809, −3.39985211422803931257754108365, −3.23993998457186077174201422279, −3.19755382976201760385207613201, −3.08255277814601535733600708402, −2.81292388530886028215583735560, −2.74085467949394956884131925052, −2.65816166632715292459209130482, −2.49857064745698613629257087570, −2.40958328255955667496056137631, −2.25936852607721271612535858736, −1.85928419422130827357233928624, −1.85914985504141915640799964743, −1.43075638842531843753231970581, −1.37623212062160312359641546678, −1.27110566387615552243386400753, −1.00967298909088852828550233605, −0.964516918168743166581048030846, −0.49287430647253588920061531901, −0.41278118102436942929658473554, −0.30830991990319752920436695867,
0.30830991990319752920436695867, 0.41278118102436942929658473554, 0.49287430647253588920061531901, 0.964516918168743166581048030846, 1.00967298909088852828550233605, 1.27110566387615552243386400753, 1.37623212062160312359641546678, 1.43075638842531843753231970581, 1.85914985504141915640799964743, 1.85928419422130827357233928624, 2.25936852607721271612535858736, 2.40958328255955667496056137631, 2.49857064745698613629257087570, 2.65816166632715292459209130482, 2.74085467949394956884131925052, 2.81292388530886028215583735560, 3.08255277814601535733600708402, 3.19755382976201760385207613201, 3.23993998457186077174201422279, 3.39985211422803931257754108365, 3.73564502301876154258415991809, 3.84416895268994950212143745180, 3.96071926360550514163616934721, 4.11386087778541214438847963591, 4.16661585298858824020409779673
Plot not available for L-functions of degree greater than 10.
