| L(s) = 1 | − 2·3-s + 4-s + 9-s + 6·11-s − 2·12-s − 4·13-s + 4·17-s + 12·25-s + 2·27-s − 6·29-s − 12·33-s + 36-s + 6·37-s + 8·39-s + 10·43-s + 6·44-s + 49-s − 8·51-s − 4·52-s + 28·53-s + 24·59-s − 64-s + 24·67-s + 4·68-s − 18·71-s − 24·75-s + 36·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 1.10·13-s + 0.970·17-s + 12/5·25-s + 0.384·27-s − 1.11·29-s − 2.08·33-s + 1/6·36-s + 0.986·37-s + 1.28·39-s + 1.52·43-s + 0.904·44-s + 1/7·49-s − 1.12·51-s − 0.554·52-s + 3.84·53-s + 3.12·59-s − 1/8·64-s + 2.93·67-s + 0.485·68-s − 2.13·71-s − 2.77·75-s + 4.05·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.616805696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.616805696\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 19 T^{2} - 4 T^{3} + 664 T^{4} - 4 p T^{5} - 19 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T + 64 T^{2} - 312 T^{3} + 1779 T^{4} - 312 p T^{5} + 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 8 T^{2} - 220 T^{3} + 5515 T^{4} - 220 p T^{5} - 8 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 162 T^{2} + 10931 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 322 T^{2} - 3120 T^{3} + 24747 T^{4} - 3120 p T^{5} + 322 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 24 T + 370 T^{2} - 4272 T^{3} + 40059 T^{4} - 4272 p T^{5} + 370 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 18 T + 268 T^{2} + 2880 T^{3} + 28227 T^{4} + 2880 p T^{5} + 268 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 27366 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 18 T + 191 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 189 T^{2} + 1692 T^{3} + 16232 T^{4} + 1692 p T^{5} + 189 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 30 T + 520 T^{2} - 6600 T^{3} + 68091 T^{4} - 6600 p T^{5} + 520 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.63007045552729697840034658127, −7.45635512146812402965429340057, −7.15873600195455352732704127888, −7.10126081181171939598290360546, −6.71584660343762529423453692116, −6.68203388395863159604548408452, −6.37243157379974201728589590801, −6.25984439166830131523684117458, −5.62999760687918158943046215073, −5.58688539823967146130806279628, −5.38567503252472026963338892338, −5.30416635511018551316427233145, −4.84680772879350861158005950088, −4.59782195039248035992870956019, −4.14794305922382447816013905362, −4.00716912079894669915291008645, −3.75889200576746739129227270549, −3.46838720640047242092413157342, −3.02685845403989968944866945151, −2.51575865112634419861401295750, −2.32096330944653758915800400891, −2.20003898206132195160429842786, −1.29360346199130214710424497254, −0.926301273700834640319346484287, −0.76537748525508855933771343786,
0.76537748525508855933771343786, 0.926301273700834640319346484287, 1.29360346199130214710424497254, 2.20003898206132195160429842786, 2.32096330944653758915800400891, 2.51575865112634419861401295750, 3.02685845403989968944866945151, 3.46838720640047242092413157342, 3.75889200576746739129227270549, 4.00716912079894669915291008645, 4.14794305922382447816013905362, 4.59782195039248035992870956019, 4.84680772879350861158005950088, 5.30416635511018551316427233145, 5.38567503252472026963338892338, 5.58688539823967146130806279628, 5.62999760687918158943046215073, 6.25984439166830131523684117458, 6.37243157379974201728589590801, 6.68203388395863159604548408452, 6.71584660343762529423453692116, 7.10126081181171939598290360546, 7.15873600195455352732704127888, 7.45635512146812402965429340057, 7.63007045552729697840034658127
