| 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
| \(L(1)\) |
\(\approx\) |
\(0.1116840141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1116840141\) |
| \(L(\frac{3}{2})\) |
|
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
−3.44722028829542747549029361082, −3.42340517050067233388037291553, −3.35893331756626149378883933572, −3.31112862158275651186014667865, −3.17246725199094981257075067431, −3.09180152778318253522117455524, −2.91142497508636568948886736646, −2.74200825224158117946595404336, −2.71579979901255716315723281247, −2.67006007725121799259722691201, −2.43889005373008141825572195038, −2.20330055326520204609258480559, −2.04424728171556886115363530564, −1.97966436534369413539468654245, −1.70322534066303850513589124078, −1.67250080550646288758600010942, −1.50573814534039013815852795211, −1.20426644388036515099877312123, −1.19846467469450061405261151503, −1.14453014272105780086333338183, −1.12575221377167456691436763035, −0.47281953770340901821317528367, −0.43173262069650757805699897322, −0.34897871331763152470663298116, −0.22696222444276819012088982547,
0.22696222444276819012088982547, 0.34897871331763152470663298116, 0.43173262069650757805699897322, 0.47281953770340901821317528367, 1.12575221377167456691436763035, 1.14453014272105780086333338183, 1.19846467469450061405261151503, 1.20426644388036515099877312123, 1.50573814534039013815852795211, 1.67250080550646288758600010942, 1.70322534066303850513589124078, 1.97966436534369413539468654245, 2.04424728171556886115363530564, 2.20330055326520204609258480559, 2.43889005373008141825572195038, 2.67006007725121799259722691201, 2.71579979901255716315723281247, 2.74200825224158117946595404336, 2.91142497508636568948886736646, 3.09180152778318253522117455524, 3.17246725199094981257075067431, 3.31112862158275651186014667865, 3.35893331756626149378883933572, 3.42340517050067233388037291553, 3.44722028829542747549029361082
Plot not available for L-functions of degree greater than 10.
