| L(s) = 1 | + 8·2-s + 36·4-s − 8·7-s + 120·8-s − 64·14-s + 330·16-s + 12·19-s + 8·25-s − 288·28-s − 16·29-s + 792·32-s + 96·38-s + 24·41-s − 32·43-s + 36·49-s + 64·50-s + 48·53-s − 960·56-s − 128·58-s + 8·61-s + 1.71e3·64-s − 16·71-s − 16·73-s + 432·76-s + 192·82-s − 256·86-s − 8·89-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s − 3.02·7-s + 42.4·8-s − 17.1·14-s + 82.5·16-s + 2.75·19-s + 8/5·25-s − 54.4·28-s − 2.97·29-s + 140.·32-s + 15.5·38-s + 3.74·41-s − 4.87·43-s + 36/7·49-s + 9.05·50-s + 6.59·53-s − 128.·56-s − 16.8·58-s + 1.02·61-s + 214.5·64-s − 1.89·71-s − 1.87·73-s + 49.5·76-s + 21.2·82-s − 27.6·86-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(882.5953733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(882.5953733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T )^{8} \) |
| 19 | \( 1 - 12 T + 44 T^{2} + 68 T^{3} - 986 T^{4} + 68 p T^{5} + 44 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( 1 - 8 T^{2} + 84 T^{4} - 504 T^{6} + 2886 T^{8} - 504 p^{2} T^{10} + 84 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 32 T^{2} + 684 T^{4} - 10944 T^{6} + 136326 T^{8} - 10944 p^{2} T^{10} + 684 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 32 T^{2} + 652 T^{4} - 10176 T^{6} + 142086 T^{8} - 10176 p^{2} T^{10} + 652 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 80 T^{2} + 3372 T^{4} - 94224 T^{6} + 1878822 T^{8} - 94224 p^{2} T^{10} + 3372 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 104 T^{2} + 5268 T^{4} - 183192 T^{6} + 4824486 T^{8} - 183192 p^{2} T^{10} + 5268 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 8 T + 100 T^{2} + 536 T^{3} + 3926 T^{4} + 536 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 160 T^{2} + 12452 T^{4} - 622816 T^{6} + 22400326 T^{8} - 622816 p^{2} T^{10} + 12452 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 144 T^{2} + 10596 T^{4} - 544304 T^{6} + 22244774 T^{8} - 544304 p^{2} T^{10} + 10596 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 12 T + 68 T^{2} - 532 T^{3} + 4806 T^{4} - 532 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 16 T + 196 T^{2} + 1632 T^{3} + 12294 T^{4} + 1632 p T^{5} + 196 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 88 T^{2} + 5212 T^{4} - 238184 T^{6} + 11562566 T^{8} - 238184 p^{2} T^{10} + 5212 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 24 T + 364 T^{2} - 3688 T^{3} + 574 p T^{4} - 3688 p T^{5} + 364 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - 4 T + 132 T^{2} - 556 T^{3} + 10806 T^{4} - 556 p T^{5} + 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 232 T^{2} + 33172 T^{4} - 3292632 T^{6} + 253066246 T^{8} - 3292632 p^{2} T^{10} + 33172 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 8 T + 220 T^{2} + 1448 T^{3} + 21926 T^{4} + 1448 p T^{5} + 220 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 164 T^{2} + 1160 T^{3} + 15766 T^{4} + 1160 p T^{5} + 164 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 272 T^{2} + 47436 T^{4} - 5706320 T^{6} + 518271078 T^{8} - 5706320 p^{2} T^{10} + 47436 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 240 T^{2} + 41348 T^{4} - 4575568 T^{6} + 438149926 T^{8} - 4575568 p^{2} T^{10} + 41348 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 4 T + 84 T^{2} + 860 T^{3} + 8422 T^{4} + 860 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 488 T^{2} + 118164 T^{4} - 18696152 T^{6} + 2119476198 T^{8} - 18696152 p^{2} T^{10} + 118164 p^{4} T^{12} - 488 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.77203904949534447832322275022, −3.68525196716369426334309966470, −3.66388139030869027203165679676, −3.36070396690343673486438633902, −3.25145137255268590413411492910, −3.15752282720635513631025011471, −3.11454144729191855180490907330, −3.09291762552602441867259191693, −2.95186727964110910430714614947, −2.80256310434919691850517195661, −2.62056850513009256418665579476, −2.61183987231393619658795783646, −2.59380860654455351872952624175, −2.28363232779957446998454783609, −2.11356616347057732663667236254, −2.01925544314924241366141534185, −1.78972541793821234473924716144, −1.70707069931503326545000747798, −1.65903250203909076752698646411, −1.41759859151982357637974255977, −1.11637436356076120596049565984, −0.920840266840014272614298634841, −0.67144243366547408297426278110, −0.56585133844857920954791116083, −0.46246273237944113459778946132,
0.46246273237944113459778946132, 0.56585133844857920954791116083, 0.67144243366547408297426278110, 0.920840266840014272614298634841, 1.11637436356076120596049565984, 1.41759859151982357637974255977, 1.65903250203909076752698646411, 1.70707069931503326545000747798, 1.78972541793821234473924716144, 2.01925544314924241366141534185, 2.11356616347057732663667236254, 2.28363232779957446998454783609, 2.59380860654455351872952624175, 2.61183987231393619658795783646, 2.62056850513009256418665579476, 2.80256310434919691850517195661, 2.95186727964110910430714614947, 3.09291762552602441867259191693, 3.11454144729191855180490907330, 3.15752282720635513631025011471, 3.25145137255268590413411492910, 3.36070396690343673486438633902, 3.66388139030869027203165679676, 3.68525196716369426334309966470, 3.77203904949534447832322275022
Plot not available for L-functions of degree greater than 10.
