| L(s) = 1 | + 2·3-s + 6·7-s + 9-s − 6·11-s − 4·17-s + 6·19-s + 12·21-s + 6·23-s + 14·25-s − 2·27-s − 2·29-s − 12·33-s + 12·37-s + 14·43-s + 8·49-s − 8·51-s − 4·53-s + 12·57-s + 16·61-s + 6·63-s + 6·67-s + 12·69-s − 18·71-s + 28·75-s − 36·77-s + 8·79-s − 4·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 1/3·9-s − 1.80·11-s − 0.970·17-s + 1.37·19-s + 2.61·21-s + 1.25·23-s + 14/5·25-s − 0.384·27-s − 0.371·29-s − 2.08·33-s + 1.97·37-s + 2.13·43-s + 8/7·49-s − 1.12·51-s − 0.549·53-s + 1.58·57-s + 2.04·61-s + 0.755·63-s + 0.733·67-s + 1.44·69-s − 2.13·71-s + 3.23·75-s − 4.10·77-s + 0.900·79-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.205854462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.205854462\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 144 T^{3} + 587 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 108 T^{3} - 573 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 7 T^{2} - 94 T^{3} - 836 T^{4} - 94 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 109 T^{2} - 732 T^{3} + 4128 T^{4} - 732 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 57 T^{2} + 1568 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 14 T + 88 T^{2} - 308 T^{3} + 1387 T^{4} - 308 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 59 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 73 T^{2} - 16 p T^{3} + 232 p T^{4} - 16 p^{2} T^{5} + 73 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 100 T^{2} - 528 T^{3} + 4059 T^{4} - 528 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 206 T^{2} + 19539 T^{4} - 206 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 842 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.504665455673927040210727836566, −8.240273894397580355119995880381, −7.87062211457166473083539087130, −7.85670532852442204936272980019, −7.68293681740288811376459245284, −7.39828777306652560919909187597, −6.97801200137537074962552513522, −6.84720356521247915658918832811, −6.63709505972233957001980735109, −6.05417870303157348316457814810, −5.68838040704008281169977780597, −5.49479501976618177001612046759, −5.26065510594354946772223595414, −4.93386284376979485275501501964, −4.64125128726007515816578639374, −4.52207681409553391408178808332, −4.34488193542714438111454445035, −3.55687267388023006281141899658, −3.40098885423237615343708825974, −2.82790479917436827897632725596, −2.75927663984589777926092830469, −2.24902829932421009545389787883, −2.18978397533391814036747931945, −1.14072775553396600938989915699, −1.10972409181099362729520177762,
1.10972409181099362729520177762, 1.14072775553396600938989915699, 2.18978397533391814036747931945, 2.24902829932421009545389787883, 2.75927663984589777926092830469, 2.82790479917436827897632725596, 3.40098885423237615343708825974, 3.55687267388023006281141899658, 4.34488193542714438111454445035, 4.52207681409553391408178808332, 4.64125128726007515816578639374, 4.93386284376979485275501501964, 5.26065510594354946772223595414, 5.49479501976618177001612046759, 5.68838040704008281169977780597, 6.05417870303157348316457814810, 6.63709505972233957001980735109, 6.84720356521247915658918832811, 6.97801200137537074962552513522, 7.39828777306652560919909187597, 7.68293681740288811376459245284, 7.85670532852442204936272980019, 7.87062211457166473083539087130, 8.240273894397580355119995880381, 8.504665455673927040210727836566
