| L(s) = 1 | − 56·7-s − 80·9-s − 144·17-s + 368·23-s − 656·25-s + 224·31-s − 368·41-s + 1.28e3·47-s + 1.76e3·49-s + 4.48e3·63-s + 256·71-s + 3.02e3·73-s + 1.85e3·79-s + 3.08e3·81-s − 16·89-s + 144·97-s + 1.05e3·103-s − 1.28e3·113-s + 8.06e3·119-s − 5.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.15e4·153-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2.96·9-s − 2.05·17-s + 3.33·23-s − 5.24·25-s + 1.29·31-s − 1.40·41-s + 3.97·47-s + 36/7·49-s + 8.95·63-s + 0.427·71-s + 4.84·73-s + 2.64·79-s + 4.23·81-s − 0.0190·89-s + 0.150·97-s + 1.01·103-s − 1.06·113-s + 6.21·119-s − 4.06·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 6.08·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.188559207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.188559207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T )^{8} \) |
| good | 3 | \( 1 + 80 T^{2} + 3316 T^{4} + 36080 p T^{6} + 3129398 T^{8} + 36080 p^{7} T^{10} + 3316 p^{12} T^{12} + 80 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 + 656 T^{2} + 213556 T^{4} + 44705744 T^{6} + 6579610742 T^{8} + 44705744 p^{6} T^{10} + 213556 p^{12} T^{12} + 656 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 + 5416 T^{2} + 12692316 T^{4} + 17331200344 T^{6} + 20605958370278 T^{8} + 17331200344 p^{6} T^{10} + 12692316 p^{12} T^{12} + 5416 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( 1 + 8624 T^{2} + 40713780 T^{4} + 129176364464 T^{6} + 318727222226486 T^{8} + 129176364464 p^{6} T^{10} + 40713780 p^{12} T^{12} + 8624 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( ( 1 + 72 T + 13628 T^{2} + 892920 T^{3} + 91744198 T^{4} + 892920 p^{3} T^{5} + 13628 p^{6} T^{6} + 72 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 + 27760 T^{2} + 407440948 T^{4} + 4211545572784 T^{6} + 33024337408498742 T^{8} + 4211545572784 p^{6} T^{10} + 407440948 p^{12} T^{12} + 27760 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 - 8 p T + 35620 T^{2} - 5050008 T^{3} + 663275414 T^{4} - 5050008 p^{3} T^{5} + 35620 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( 1 + 143304 T^{2} + 9930387964 T^{4} + 428425053590712 T^{6} + 12545170807824460390 T^{8} + 428425053590712 p^{6} T^{10} + 9930387964 p^{12} T^{12} + 143304 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 - 112 T + 84124 T^{2} - 8974000 T^{3} + 3237184838 T^{4} - 8974000 p^{3} T^{5} + 84124 p^{6} T^{6} - 112 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 + 186120 T^{2} + 16942615612 T^{4} + 1111623636531192 T^{6} + 61133206592686517926 T^{8} + 1111623636531192 p^{6} T^{10} + 16942615612 p^{12} T^{12} + 186120 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 + 184 T + 201532 T^{2} + 31893960 T^{3} + 18186112358 T^{4} + 31893960 p^{3} T^{5} + 201532 p^{6} T^{6} + 184 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 + 255528 T^{2} + 30692831836 T^{4} + 2169386962708056 T^{6} + \)\(14\!\cdots\!90\)\( T^{8} + 2169386962708056 p^{6} T^{10} + 30692831836 p^{12} T^{12} + 255528 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 - 640 T + 258492 T^{2} - 114135168 T^{3} + 47609965638 T^{4} - 114135168 p^{3} T^{5} + 258492 p^{6} T^{6} - 640 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( 1 + 776296 T^{2} + 304939644540 T^{4} + 77243997567119512 T^{6} + \)\(13\!\cdots\!50\)\( T^{8} + 77243997567119512 p^{6} T^{10} + 304939644540 p^{12} T^{12} + 776296 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 + 755184 T^{2} + 215895789684 T^{4} + 27977406392199728 T^{6} + \)\(28\!\cdots\!38\)\( T^{8} + 27977406392199728 p^{6} T^{10} + 215895789684 p^{12} T^{12} + 755184 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( 1 + 1146000 T^{2} + 614059813876 T^{4} + 209582944055251920 T^{6} + \)\(53\!\cdots\!38\)\( T^{8} + 209582944055251920 p^{6} T^{10} + 614059813876 p^{12} T^{12} + 1146000 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( 1 + 317896 T^{2} + 122917913052 T^{4} + 42308127361031224 T^{6} + \)\(58\!\cdots\!58\)\( T^{8} + 42308127361031224 p^{6} T^{10} + 122917913052 p^{12} T^{12} + 317896 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 128 T + 552124 T^{2} + 118565760 T^{3} + 195798414758 T^{4} + 118565760 p^{3} T^{5} + 552124 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 1512 T + 1561628 T^{2} - 910368024 T^{3} + 590001691430 T^{4} - 910368024 p^{3} T^{5} + 1561628 p^{6} T^{6} - 1512 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 928 T + 1281564 T^{2} - 935644448 T^{3} + 779746151942 T^{4} - 935644448 p^{3} T^{5} + 1281564 p^{6} T^{6} - 928 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 + 3544528 T^{2} + 5853189245172 T^{4} + 5936956701665785168 T^{6} + \)\(40\!\cdots\!70\)\( T^{8} + 5936956701665785168 p^{6} T^{10} + 5853189245172 p^{12} T^{12} + 3544528 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 + 8 T + 1253404 T^{2} - 603300232 T^{3} + 723791952102 T^{4} - 603300232 p^{3} T^{5} + 1253404 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 72 T + 2589180 T^{2} - 248791672 T^{3} + 3133709256966 T^{4} - 248791672 p^{3} T^{5} + 2589180 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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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
−3.57665564811182593020730335090, −3.20646232235399954348111624037, −3.08897423889994843300644769998, −3.08248169161315792632337230938, −3.01204612845905749192792291014, −2.94761259557091964168096381437, −2.87326282593005274836605286747, −2.76065277830939745502135264396, −2.47650992978851721180199588727, −2.29644399368397106449161256737, −2.21401527988169887893311133637, −2.17228602981777137537826668507, −2.07923362591752218012421658252, −1.96446852729174320399968472194, −1.82153089854935312612527981265, −1.51601858871081690492153729101, −1.45673180488987831711972721088, −1.06624173879420060619591133175, −0.72400777002689619585230063641, −0.71051484717152700424000562922, −0.70134256228582610271003360718, −0.52287151559445961332406438527, −0.42965160889924950136067097503, −0.28771974664581841079038876447, −0.22098977754913101569837264791,
0.22098977754913101569837264791, 0.28771974664581841079038876447, 0.42965160889924950136067097503, 0.52287151559445961332406438527, 0.70134256228582610271003360718, 0.71051484717152700424000562922, 0.72400777002689619585230063641, 1.06624173879420060619591133175, 1.45673180488987831711972721088, 1.51601858871081690492153729101, 1.82153089854935312612527981265, 1.96446852729174320399968472194, 2.07923362591752218012421658252, 2.17228602981777137537826668507, 2.21401527988169887893311133637, 2.29644399368397106449161256737, 2.47650992978851721180199588727, 2.76065277830939745502135264396, 2.87326282593005274836605286747, 2.94761259557091964168096381437, 3.01204612845905749192792291014, 3.08248169161315792632337230938, 3.08897423889994843300644769998, 3.20646232235399954348111624037, 3.57665564811182593020730335090
Plot not available for L-functions of degree greater than 10.
