| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s + 330·16-s + 16·25-s − 792·32-s + 64·43-s − 4·49-s − 128·50-s + 1.71e3·64-s + 32·71-s − 48·79-s + 10·81-s − 512·86-s + 32·98-s + 576·100-s − 88·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + 139-s − 256·142-s + 149-s + 151-s + 157-s + 384·158-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s + 82.5·16-s + 16/5·25-s − 140.·32-s + 9.75·43-s − 4/7·49-s − 18.1·50-s + 214.5·64-s + 3.79·71-s − 5.40·79-s + 10/9·81-s − 55.2·86-s + 3.23·98-s + 57.5·100-s − 8·121-s + 0.0887·127-s − 303.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 21.4·142-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 30.5·158-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2998272401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2998272401\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 10 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 + 4 T^{2} - 10 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | \( 1 + 310 T^{4} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 - 8 T^{2} + 38 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 12 T^{2} + 166 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 24 T^{2} + 838 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 28 T^{2} + 246 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 84 T^{2} + 3334 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 84 T^{2} + 3574 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 116 T^{2} + 5990 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 52 T^{2} + 3926 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 108 T^{2} + 6886 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 148 T^{2} + 10086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 216 T^{2} + 18598 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 192 T^{2} + 16630 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 92 T^{2} + 3926 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 124 T^{2} + 9014 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 152 T^{2} + 18182 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 188 T^{2} + 19190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 172 T^{2} + 26102 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.79110417050554257212035033348, −4.60813431425811412251450161687, −4.51935334607244410788646319847, −4.08828198338155360025778847272, −4.02012577127890003000260425762, −3.94724809318322377748231286009, −3.83278288394319575204873458359, −3.78491915492665488540176545446, −3.40214944337826470161130702305, −2.95352209525819191065984164038, −2.92023749726845299461940628308, −2.91159062099861472238650177348, −2.76493024661493328844778085886, −2.67186854168807657767385234989, −2.55179049973386911530200234851, −2.20387788111732723482299890587, −2.10854054547831715185297251060, −2.06149969785040731582345893104, −1.69696089241618533584007584827, −1.26197729513931330076078937803, −1.23108914369029048240047038509, −1.05062880503702749345095537488, −0.915252312202299965815811968374, −0.65132659530273040693232371101, −0.36000312018704928925885794995,
0.36000312018704928925885794995, 0.65132659530273040693232371101, 0.915252312202299965815811968374, 1.05062880503702749345095537488, 1.23108914369029048240047038509, 1.26197729513931330076078937803, 1.69696089241618533584007584827, 2.06149969785040731582345893104, 2.10854054547831715185297251060, 2.20387788111732723482299890587, 2.55179049973386911530200234851, 2.67186854168807657767385234989, 2.76493024661493328844778085886, 2.91159062099861472238650177348, 2.92023749726845299461940628308, 2.95352209525819191065984164038, 3.40214944337826470161130702305, 3.78491915492665488540176545446, 3.83278288394319575204873458359, 3.94724809318322377748231286009, 4.02012577127890003000260425762, 4.08828198338155360025778847272, 4.51935334607244410788646319847, 4.60813431425811412251450161687, 4.79110417050554257212035033348
Plot not available for L-functions of degree greater than 10.
