| L(s) = 1 | + 2·3-s + 8·5-s + 16·15-s − 8·19-s − 12·23-s + 36·25-s + 2·27-s − 8·29-s − 36·43-s + 4·47-s + 28·49-s − 16·57-s − 28·67-s − 24·69-s − 24·71-s + 32·73-s + 72·75-s + 10·81-s − 16·87-s − 64·95-s + 16·97-s + 8·101-s − 96·115-s + 24·121-s + 120·125-s + 127-s − 72·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3.57·5-s + 4.13·15-s − 1.83·19-s − 2.50·23-s + 36/5·25-s + 0.384·27-s − 1.48·29-s − 5.48·43-s + 0.583·47-s + 4·49-s − 2.11·57-s − 3.42·67-s − 2.88·69-s − 2.84·71-s + 3.74·73-s + 8.31·75-s + 10/9·81-s − 1.71·87-s − 6.56·95-s + 1.62·97-s + 0.796·101-s − 8.95·115-s + 2.18·121-s + 10.7·125-s + 0.0887·127-s − 6.33·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.346712941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.346712941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 4 T^{2} - 10 T^{3} + 14 T^{4} - 10 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 - T )^{8} \) |
| good | 7 | \( 1 - 4 p T^{2} + 388 T^{4} - 3684 T^{6} + 28022 T^{8} - 3684 p^{2} T^{10} + 388 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 24 T^{2} + 316 T^{4} - 3944 T^{6} + 48998 T^{8} - 3944 p^{2} T^{10} + 316 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 56 T^{2} + 1724 T^{4} - 35528 T^{6} + 536358 T^{8} - 35528 p^{2} T^{10} + 1724 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 72 T^{2} + 2556 T^{4} - 63864 T^{6} + 1234694 T^{8} - 63864 p^{2} T^{10} + 2556 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 4 T + 36 T^{2} + 212 T^{3} + 806 T^{4} + 212 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 6 T + 68 T^{2} + 206 T^{3} + 1742 T^{4} + 206 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 52 T^{2} - 52 T^{3} + 742 T^{4} - 52 p T^{5} + 52 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 168 T^{2} + 12796 T^{4} - 608536 T^{6} + 21299078 T^{8} - 608536 p^{2} T^{10} + 12796 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 88 T^{2} + 6716 T^{4} - 333608 T^{6} + 14506726 T^{8} - 333608 p^{2} T^{10} + 6716 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 104 T^{2} + 8732 T^{4} - 458584 T^{6} + 21948550 T^{8} - 458584 p^{2} T^{10} + 8732 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 18 T + 228 T^{2} + 2106 T^{3} + 15918 T^{4} + 2106 p T^{5} + 228 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2 T + 132 T^{2} - 346 T^{3} + 8046 T^{4} - 346 p T^{5} + 132 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 124 T^{2} + 192 T^{3} + 7990 T^{4} + 192 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 296 T^{2} + 40156 T^{4} - 3466968 T^{6} + 226889894 T^{8} - 3466968 p^{2} T^{10} + 40156 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 200 T^{2} + 19644 T^{4} - 1146424 T^{6} + 62381222 T^{8} - 1146424 p^{2} T^{10} + 19644 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 14 T + 276 T^{2} + 2518 T^{3} + 28206 T^{4} + 2518 p T^{5} + 276 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 12 T + 156 T^{2} + 764 T^{3} + 7334 T^{4} + 764 p T^{5} + 156 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 16 T + 204 T^{2} - 1712 T^{3} + 16262 T^{4} - 1712 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 280 T^{2} + 45020 T^{4} - 5280936 T^{6} + 474214470 T^{8} - 5280936 p^{2} T^{10} + 45020 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 476 T^{2} + 108228 T^{4} - 15423172 T^{6} + 1519238486 T^{8} - 15423172 p^{2} T^{10} + 108228 p^{4} T^{12} - 476 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 520 T^{2} + 131036 T^{4} - 20628664 T^{6} + 2203778694 T^{8} - 20628664 p^{2} T^{10} + 131036 p^{4} T^{12} - 520 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 8 T + 316 T^{2} - 1848 T^{3} + 43142 T^{4} - 1848 p T^{5} + 316 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.96050023757798342066342495331, −3.74713487752559348570540770770, −3.41741608443812904109404691297, −3.40127115573528978008925558359, −3.30016374721936220056994923592, −3.13462557778124715104195542668, −3.12458788851363720378505058126, −3.11033727986350076406636283310, −2.86092403170146078610526271186, −2.59949082965905443143750987374, −2.40140500752461640595067104638, −2.34565609894506034640840256000, −2.25757415493718466189164079050, −2.23703315733247154654460840796, −2.11780484868781863952851613563, −2.00174121998562996892552715739, −1.78067825568675621843216616909, −1.62159311825382029670900475993, −1.53525577186135650464752646506, −1.38012991765055483398356695935, −1.36590757348107049174029051091, −0.901767947883607964217380070338, −0.74516938252246295847371292070, −0.39751318164959398993067791962, −0.16987191105853134426170722587,
0.16987191105853134426170722587, 0.39751318164959398993067791962, 0.74516938252246295847371292070, 0.901767947883607964217380070338, 1.36590757348107049174029051091, 1.38012991765055483398356695935, 1.53525577186135650464752646506, 1.62159311825382029670900475993, 1.78067825568675621843216616909, 2.00174121998562996892552715739, 2.11780484868781863952851613563, 2.23703315733247154654460840796, 2.25757415493718466189164079050, 2.34565609894506034640840256000, 2.40140500752461640595067104638, 2.59949082965905443143750987374, 2.86092403170146078610526271186, 3.11033727986350076406636283310, 3.12458788851363720378505058126, 3.13462557778124715104195542668, 3.30016374721936220056994923592, 3.40127115573528978008925558359, 3.41741608443812904109404691297, 3.74713487752559348570540770770, 3.96050023757798342066342495331
Plot not available for L-functions of degree greater than 10.
