| L(s) = 1 | + 4·3-s − 2·4-s + 10·9-s − 8·12-s + 6·13-s + 3·16-s + 8·17-s − 4·23-s − 2·25-s + 20·27-s + 8·29-s − 20·36-s + 24·39-s − 8·43-s + 12·48-s − 8·49-s + 32·51-s − 12·52-s − 4·53-s + 40·61-s − 4·64-s − 16·68-s − 16·69-s − 8·75-s + 32·79-s + 35·81-s + 32·87-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 1.66·13-s + 3/4·16-s + 1.94·17-s − 0.834·23-s − 2/5·25-s + 3.84·27-s + 1.48·29-s − 3.33·36-s + 3.84·39-s − 1.21·43-s + 1.73·48-s − 8/7·49-s + 4.48·51-s − 1.66·52-s − 0.549·53-s + 5.12·61-s − 1/2·64-s − 1.94·68-s − 1.92·69-s − 0.923·75-s + 3.60·79-s + 35/9·81-s + 3.43·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.692411814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.692411814\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 46 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 3262 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 3814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 14814 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 4766 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13894 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 7454 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 21526 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 22526 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.122968279003970163132267309979, −8.072100298636340252821930282454, −8.022360486362146270906906186783, −7.80969928564689767813418066144, −7.34746814183212372238007293265, −6.91726724913634033252255118654, −6.64359584280747701279743440924, −6.56608982688036452895534213985, −6.49499788494832839476608606612, −5.70088274447625252525906654435, −5.59683933014197947420380552559, −5.36126612194867428113126943173, −5.10594149435196506632106662911, −4.71844180286973510543996675000, −4.17139879793489547440662405322, −4.14944062459346395365861817459, −3.82735430666502852902875426281, −3.47828406412126673822702117232, −3.45992713194320431476358006364, −2.87150561423351129406409523583, −2.83995846023222531482386603290, −2.05518018277053265444179179702, −1.94216972204415676821100824926, −1.12154252455089545342960236453, −1.04359650279217571606764863877,
1.04359650279217571606764863877, 1.12154252455089545342960236453, 1.94216972204415676821100824926, 2.05518018277053265444179179702, 2.83995846023222531482386603290, 2.87150561423351129406409523583, 3.45992713194320431476358006364, 3.47828406412126673822702117232, 3.82735430666502852902875426281, 4.14944062459346395365861817459, 4.17139879793489547440662405322, 4.71844180286973510543996675000, 5.10594149435196506632106662911, 5.36126612194867428113126943173, 5.59683933014197947420380552559, 5.70088274447625252525906654435, 6.49499788494832839476608606612, 6.56608982688036452895534213985, 6.64359584280747701279743440924, 6.91726724913634033252255118654, 7.34746814183212372238007293265, 7.80969928564689767813418066144, 8.022360486362146270906906186783, 8.072100298636340252821930282454, 8.122968279003970163132267309979
