| L(s) = 1 | + 2·3-s − 4·7-s − 9-s + 8·11-s − 8·13-s + 10·17-s − 8·21-s + 14·23-s − 2·27-s + 18·29-s + 16·31-s + 16·33-s + 4·37-s − 16·39-s − 4·41-s + 26·43-s + 16·47-s + 15·49-s + 20·51-s + 20·53-s − 24·59-s + 2·61-s + 4·63-s − 12·67-s + 28·69-s + 8·71-s − 32·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s − 1/3·9-s + 2.41·11-s − 2.21·13-s + 2.42·17-s − 1.74·21-s + 2.91·23-s − 0.384·27-s + 3.34·29-s + 2.87·31-s + 2.78·33-s + 0.657·37-s − 2.56·39-s − 0.624·41-s + 3.96·43-s + 2.33·47-s + 15/7·49-s + 2.80·51-s + 2.74·53-s − 3.12·59-s + 0.256·61-s + 0.503·63-s − 1.46·67-s + 3.37·69-s + 0.949·71-s − 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.824246041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.824246041\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + T^{2} + 4 T^{3} + 64 T^{4} + 4 p T^{5} + p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 8 p T^{3} - 52 p T^{4} + 8 p^{2} T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 70 T^{3} + 16 T^{4} - 70 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 96 T^{3} - 124 T^{4} - 96 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 226 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 18 T + 177 T^{2} - 1242 T^{3} + 7052 T^{4} - 1242 p T^{5} + 177 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 912 T^{3} + 5822 T^{4} - 912 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 59 T^{2} - 4 T^{3} + 3664 T^{4} - 4 p T^{5} - 59 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 424 T^{3} + 2092 T^{4} + 424 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 26 T + 365 T^{2} - 3474 T^{3} + 25520 T^{4} - 3474 p T^{5} + 365 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1580 T^{3} + 11806 T^{4} - 1580 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 153 T^{2} - 1176 T^{3} - 21292 T^{4} - 1176 p T^{5} + 153 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 113 T^{2} + 780 T^{3} + 2952 T^{4} + 780 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 272 T^{3} - 1748 T^{4} - 272 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 16134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 28 T + 197 T^{2} - 2408 T^{3} - 48788 T^{4} - 2408 p T^{5} + 197 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 253 T^{2} + 2460 T^{3} + 37272 T^{4} + 2460 p T^{5} + 253 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.05155513399885368193349374477, −6.65864657314150596322836889650, −6.42443340858370305290149878451, −6.35517386263388108493073358760, −6.14244634781796097174162653011, −5.78563898306963835035772386326, −5.57741726709178369379746829616, −5.38958447625028017282321185429, −5.23844381968135501393929709124, −4.58424973762376389301100943033, −4.48100521113412038250289475827, −4.45297523352078637322337427233, −4.33522995104791983786303262190, −3.76540583735124761867465640936, −3.60875228488871049610477177898, −3.23204850639529426093298874515, −2.94324748014175952691954411500, −2.86349475675862483809855310402, −2.62723058252487468202790168619, −2.55973536191366416040470284538, −2.38060152960421022031031006421, −1.35921956322910521343344334389, −1.03190059213030764560644618192, −0.961448248374369425102248000128, −0.74334641785042497390203144275,
0.74334641785042497390203144275, 0.961448248374369425102248000128, 1.03190059213030764560644618192, 1.35921956322910521343344334389, 2.38060152960421022031031006421, 2.55973536191366416040470284538, 2.62723058252487468202790168619, 2.86349475675862483809855310402, 2.94324748014175952691954411500, 3.23204850639529426093298874515, 3.60875228488871049610477177898, 3.76540583735124761867465640936, 4.33522995104791983786303262190, 4.45297523352078637322337427233, 4.48100521113412038250289475827, 4.58424973762376389301100943033, 5.23844381968135501393929709124, 5.38958447625028017282321185429, 5.57741726709178369379746829616, 5.78563898306963835035772386326, 6.14244634781796097174162653011, 6.35517386263388108493073358760, 6.42443340858370305290149878451, 6.65864657314150596322836889650, 7.05155513399885368193349374477
