L(s) = 1 | − 4-s − 6·7-s + 8-s − 3·11-s − 6·13-s + 13·17-s + 11·19-s + 13·23-s + 6·28-s + 3·29-s − 11·31-s + 32-s − 21·37-s + 41-s − 2·43-s + 3·44-s + 14·47-s − 10·49-s + 6·52-s + 23·53-s − 6·56-s − 9·59-s + 11·61-s − 9·64-s − 8·67-s − 13·68-s + 8·71-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 2.26·7-s + 0.353·8-s − 0.904·11-s − 1.66·13-s + 3.15·17-s + 2.52·19-s + 2.71·23-s + 1.13·28-s + 0.557·29-s − 1.97·31-s + 0.176·32-s − 3.45·37-s + 0.156·41-s − 0.304·43-s + 0.452·44-s + 2.04·47-s − 1.42·49-s + 0.832·52-s + 3.15·53-s − 0.801·56-s − 1.17·59-s + 1.40·61-s − 9/8·64-s − 0.977·67-s − 1.57·68-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.870268462\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.870268462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + T^{2} - T^{3} + T^{4} - 3 T^{5} + 11 T^{6} - 3 p T^{7} + p^{2} T^{8} - p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 6 T + 46 T^{2} + 185 T^{3} + 822 T^{4} + 2435 T^{5} + 7706 T^{6} + 2435 p T^{7} + 822 p^{2} T^{8} + 185 p^{3} T^{9} + 46 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 3 T + 42 T^{2} + 9 p T^{3} + 870 T^{4} + 1842 T^{5} + 11882 T^{6} + 1842 p T^{7} + 870 p^{2} T^{8} + 9 p^{4} T^{9} + 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 6 T + 4 p T^{2} + 217 T^{3} + 1284 T^{4} + 4378 T^{5} + 20303 T^{6} + 4378 p T^{7} + 1284 p^{2} T^{8} + 217 p^{3} T^{9} + 4 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 13 T + 117 T^{2} - 40 p T^{3} + 3545 T^{4} - 15273 T^{5} + 67369 T^{6} - 15273 p T^{7} + 3545 p^{2} T^{8} - 40 p^{4} T^{9} + 117 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 11 T + 125 T^{2} - 876 T^{3} + 6052 T^{4} - 30827 T^{5} + 424 p^{2} T^{6} - 30827 p T^{7} + 6052 p^{2} T^{8} - 876 p^{3} T^{9} + 125 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 13 T + 144 T^{2} - 1185 T^{3} + 364 p T^{4} - 49500 T^{5} + 255074 T^{6} - 49500 p T^{7} + 364 p^{3} T^{8} - 1185 p^{3} T^{9} + 144 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 3 T + 133 T^{2} - 262 T^{3} + 8085 T^{4} - 12009 T^{5} + 296107 T^{6} - 12009 p T^{7} + 8085 p^{2} T^{8} - 262 p^{3} T^{9} + 133 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 11 T + 140 T^{2} + 1025 T^{3} + 8226 T^{4} + 46520 T^{5} + 297614 T^{6} + 46520 p T^{7} + 8226 p^{2} T^{8} + 1025 p^{3} T^{9} + 140 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 21 T + 361 T^{2} + 4000 T^{3} + 38967 T^{4} + 7915 p T^{5} + 1990421 T^{6} + 7915 p^{2} T^{7} + 38967 p^{2} T^{8} + 4000 p^{3} T^{9} + 361 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - T + 65 T^{2} - 315 T^{3} + 4526 T^{4} - 16885 T^{5} + 207789 T^{6} - 16885 p T^{7} + 4526 p^{2} T^{8} - 315 p^{3} T^{9} + 65 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 2 T + 167 T^{2} + 256 T^{3} + 13452 T^{4} + 15956 T^{5} + 692036 T^{6} + 15956 p T^{7} + 13452 p^{2} T^{8} + 256 p^{3} T^{9} + 167 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 14 T + 278 T^{2} - 2603 T^{3} + 30072 T^{4} - 211487 T^{5} + 1808494 T^{6} - 211487 p T^{7} + 30072 p^{2} T^{8} - 2603 p^{3} T^{9} + 278 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 23 T + 379 T^{2} - 4310 T^{3} + 40637 T^{4} - 323685 T^{5} + 2441559 T^{6} - 323685 p T^{7} + 40637 p^{2} T^{8} - 4310 p^{3} T^{9} + 379 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 9 T + 265 T^{2} + 2264 T^{3} + 32442 T^{4} + 249123 T^{5} + 2397904 T^{6} + 249123 p T^{7} + 32442 p^{2} T^{8} + 2264 p^{3} T^{9} + 265 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 11 T + 227 T^{2} - 1507 T^{3} + 23288 T^{4} - 138147 T^{5} + 1774033 T^{6} - 138147 p T^{7} + 23288 p^{2} T^{8} - 1507 p^{3} T^{9} + 227 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 8 T + 247 T^{2} + 1290 T^{3} + 26550 T^{4} + 92768 T^{5} + 1956344 T^{6} + 92768 p T^{7} + 26550 p^{2} T^{8} + 1290 p^{3} T^{9} + 247 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 8 T + 276 T^{2} - 1095 T^{3} + 27860 T^{4} - 28473 T^{5} + 1889114 T^{6} - 28473 p T^{7} + 27860 p^{2} T^{8} - 1095 p^{3} T^{9} + 276 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 13 T + 364 T^{2} + 3570 T^{3} + 58057 T^{4} + 450270 T^{5} + 5397049 T^{6} + 450270 p T^{7} + 58057 p^{2} T^{8} + 3570 p^{3} T^{9} + 364 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 5 T + 314 T^{2} + 1445 T^{3} + 45010 T^{4} + 190240 T^{5} + 4170310 T^{6} + 190240 p T^{7} + 45010 p^{2} T^{8} + 1445 p^{3} T^{9} + 314 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 20 T + 524 T^{2} + 7048 T^{3} + 106936 T^{4} + 1073796 T^{5} + 11716314 T^{6} + 1073796 p T^{7} + 106936 p^{2} T^{8} + 7048 p^{3} T^{9} + 524 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 4 T + 70 T^{2} - 1649 T^{3} + 9072 T^{4} - 84358 T^{5} + 1538759 T^{6} - 84358 p T^{7} + 9072 p^{2} T^{8} - 1649 p^{3} T^{9} + 70 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 7 T + 367 T^{2} - 1905 T^{3} + 69210 T^{4} - 304027 T^{5} + 8348279 T^{6} - 304027 p T^{7} + 69210 p^{2} T^{8} - 1905 p^{3} T^{9} + 367 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.35517004591866126745666461012, −3.78126267251125159436491386682, −3.70627129060984736909146602527, −3.69879737679564379824805581532, −3.56718245557094863501175449574, −3.33090581184449465476306446440, −3.29261104145051872889558258509, −3.21610097849023446568333709148, −3.11563733635217610049025414612, −2.95170825564703598804916283713, −2.84718463169421640630631421969, −2.72613144922462233325021435196, −2.60041485805015664536286515145, −2.36242749181773060006634961586, −2.17162573620598780266860342356, −1.87317647987066047457216607159, −1.60324013865665778616347200561, −1.58269953710054108240292935980, −1.54477262692231714187077074971, −1.30889348925873739567633997416, −0.893698584710080695866631914176, −0.801452305009849189091906986947, −0.51909783592624251110818563400, −0.39120460119267850683692091073, −0.36046574005908201012212072555,
0.36046574005908201012212072555, 0.39120460119267850683692091073, 0.51909783592624251110818563400, 0.801452305009849189091906986947, 0.893698584710080695866631914176, 1.30889348925873739567633997416, 1.54477262692231714187077074971, 1.58269953710054108240292935980, 1.60324013865665778616347200561, 1.87317647987066047457216607159, 2.17162573620598780266860342356, 2.36242749181773060006634961586, 2.60041485805015664536286515145, 2.72613144922462233325021435196, 2.84718463169421640630631421969, 2.95170825564703598804916283713, 3.11563733635217610049025414612, 3.21610097849023446568333709148, 3.29261104145051872889558258509, 3.33090581184449465476306446440, 3.56718245557094863501175449574, 3.69879737679564379824805581532, 3.70627129060984736909146602527, 3.78126267251125159436491386682, 4.35517004591866126745666461012
Plot not available for L-functions of degree greater than 10.