| L(s) = 1 | + 2·2-s + 2·4-s − 12·7-s + 4·8-s + 4·9-s − 24·14-s + 8·16-s + 8·17-s + 8·18-s + 16·23-s − 4·25-s − 24·28-s − 20·31-s + 8·32-s + 16·34-s + 8·36-s + 8·41-s + 32·46-s + 20·47-s + 68·49-s − 8·50-s − 48·56-s − 40·62-s − 48·63-s + 8·64-s + 16·68-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 4.53·7-s + 1.41·8-s + 4/3·9-s − 6.41·14-s + 2·16-s + 1.94·17-s + 1.88·18-s + 3.33·23-s − 4/5·25-s − 4.53·28-s − 3.59·31-s + 1.41·32-s + 2.74·34-s + 4/3·36-s + 1.24·41-s + 4.71·46-s + 2.91·47-s + 68/7·49-s − 1.13·50-s − 6.41·56-s − 5.08·62-s − 6.04·63-s + 64-s + 1.94·68-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582913568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582913568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8618 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 25866 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 276 T^{2} + 32522 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.975598714293025556706726442622, −9.927458358212439716120657493433, −9.789178435230406437411541607628, −9.514710332505391917145492678906, −9.093944912592433757917183976042, −8.999722199462404836567638118216, −8.769027114441600982185064524032, −7.88710400559734548488988068574, −7.39930033665639993270289510149, −7.37029897698755686362765248658, −7.15368383405723015950202175585, −6.85944620926091183156489416071, −6.75796848113682246252332617157, −6.03341198059096965809599785503, −5.76050998433423880724887136900, −5.65706916745960590564750799286, −5.50205270731119407441896095195, −4.56746430097494365243897227587, −4.48177866106527543337411071211, −3.75203968154316350580012571200, −3.56327089894437307225813162307, −3.35667813713256326653348392359, −3.06320050117788563455232785257, −2.48106307219566836217102578034, −1.23000020128735635730991750156,
1.23000020128735635730991750156, 2.48106307219566836217102578034, 3.06320050117788563455232785257, 3.35667813713256326653348392359, 3.56327089894437307225813162307, 3.75203968154316350580012571200, 4.48177866106527543337411071211, 4.56746430097494365243897227587, 5.50205270731119407441896095195, 5.65706916745960590564750799286, 5.76050998433423880724887136900, 6.03341198059096965809599785503, 6.75796848113682246252332617157, 6.85944620926091183156489416071, 7.15368383405723015950202175585, 7.37029897698755686362765248658, 7.39930033665639993270289510149, 7.88710400559734548488988068574, 8.769027114441600982185064524032, 8.999722199462404836567638118216, 9.093944912592433757917183976042, 9.514710332505391917145492678906, 9.789178435230406437411541607628, 9.927458358212439716120657493433, 9.975598714293025556706726442622
