| L(s) = 1 | + 2·3-s − 4·5-s − 6·7-s − 2·9-s − 8·15-s + 6·17-s − 12·21-s − 6·23-s + 10·25-s − 8·27-s + 24·35-s − 18·37-s − 24·41-s + 10·43-s + 8·45-s + 2·49-s + 12·51-s + 12·53-s − 12·59-s + 4·61-s + 12·63-s − 18·67-s − 12·69-s − 12·71-s − 24·73-s + 20·75-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 2.26·7-s − 2/3·9-s − 2.06·15-s + 1.45·17-s − 2.61·21-s − 1.25·23-s + 2·25-s − 1.53·27-s + 4.05·35-s − 2.95·37-s − 3.74·41-s + 1.52·43-s + 1.19·45-s + 2/7·49-s + 1.68·51-s + 1.64·53-s − 1.56·59-s + 0.512·61-s + 1.51·63-s − 2.19·67-s − 1.44·69-s − 1.42·71-s − 2.80·73-s + 2.30·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 2 p T^{2} - 8 T^{3} + 19 T^{4} - 8 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 34 T^{2} + 120 T^{3} + 375 T^{4} + 120 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 50 T^{2} - 252 T^{3} + 1095 T^{4} - 252 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 46 T^{2} + 1059 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 74 T^{2} + 384 T^{3} + 2343 T^{4} + 384 p T^{5} + 74 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 74 T^{2} - 96 T^{3} + 2571 T^{4} - 96 p T^{5} + 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 28 T^{2} + 1926 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 18 T + 214 T^{2} + 1836 T^{3} + 12891 T^{4} + 1836 p T^{5} + 214 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 106 T^{2} - 376 T^{3} + 3175 T^{4} - 376 p T^{5} + 106 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 56 T^{2} + 12 T^{3} - 306 T^{4} + 12 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 194 T^{2} + 1512 T^{3} + 15183 T^{4} + 1512 p T^{5} + 194 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 142 T^{2} - 88 T^{3} + 8911 T^{4} - 88 p T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 18 T + 262 T^{2} + 2808 T^{3} + 27171 T^{4} + 2808 p T^{5} + 262 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 122 T^{2} + 72 T^{3} + 519 T^{4} + 72 p T^{5} + 122 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 448 T^{2} + 72 p T^{3} + 53166 T^{4} + 72 p^{2} T^{5} + 448 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 136 T^{2} + 392 T^{3} + 8638 T^{4} + 392 p T^{5} + 136 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 36 T + 740 T^{2} - 10404 T^{3} + 109638 T^{4} - 10404 p T^{5} + 740 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 314 T^{2} + 2880 T^{3} + 40167 T^{4} + 2880 p T^{5} + 314 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 202 T^{2} - 84 T^{3} + 16767 T^{4} - 84 p T^{5} + 202 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.61504806764341990715151267165, −6.17091324936947484954216458817, −6.03575085915405888433145720416, −5.98825685620925884628086607923, −5.91989813057228265614005165248, −5.37949231357283270700091924865, −5.26812447162241826503392967940, −5.07021125997773992490662395691, −5.01219873359980723896000849875, −4.56965377143599897923992644979, −4.43223815682522007596046005879, −3.91817992401870141048385712586, −3.90777818267459454652746359477, −3.76661470384509717189939341172, −3.56857269136967380050693133927, −3.31166485769369421580125833000, −3.25373300523161161491786613738, −3.00828967934133631772627827094, −2.80040856887852780487014891066, −2.56768079940310418300316069890, −2.48906461017741850908403747312, −1.89330756471848981801296664051, −1.52533310421739691100628118361, −1.38062297782830963989479675054, −1.12585819767334998460404059314, 0, 0, 0, 0,
1.12585819767334998460404059314, 1.38062297782830963989479675054, 1.52533310421739691100628118361, 1.89330756471848981801296664051, 2.48906461017741850908403747312, 2.56768079940310418300316069890, 2.80040856887852780487014891066, 3.00828967934133631772627827094, 3.25373300523161161491786613738, 3.31166485769369421580125833000, 3.56857269136967380050693133927, 3.76661470384509717189939341172, 3.90777818267459454652746359477, 3.91817992401870141048385712586, 4.43223815682522007596046005879, 4.56965377143599897923992644979, 5.01219873359980723896000849875, 5.07021125997773992490662395691, 5.26812447162241826503392967940, 5.37949231357283270700091924865, 5.91989813057228265614005165248, 5.98825685620925884628086607923, 6.03575085915405888433145720416, 6.17091324936947484954216458817, 6.61504806764341990715151267165
