| L(s) = 1 | + 1.19e3·4-s + 1.09e5·11-s + 2.77e5·16-s − 1.63e6·19-s + 4.35e6·29-s + 8.54e6·31-s − 1.18e7·41-s + 1.30e8·44-s + 1.27e8·49-s + 1.13e7·59-s + 2.50e8·61-s − 4.12e8·64-s − 5.95e8·71-s − 1.95e9·76-s + 6.20e8·79-s − 2.20e9·89-s + 9.16e8·101-s − 4.25e9·109-s + 5.19e9·116-s − 2.65e9·121-s + 1.02e10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.33·4-s + 2.25·11-s + 1.05·16-s − 2.88·19-s + 1.14·29-s + 1.66·31-s − 0.655·41-s + 5.25·44-s + 3.15·49-s + 0.121·59-s + 2.31·61-s − 3.07·64-s − 2.77·71-s − 6.72·76-s + 1.79·79-s − 3.72·89-s + 0.876·101-s − 2.88·109-s + 2.66·116-s − 1.12·121-s + 3.88·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.251396271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.251396271\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 1195 T^{2} + 287733 p^{2} T^{4} - 9864815 p^{6} T^{6} + 287733 p^{20} T^{8} - 1195 p^{36} T^{10} + p^{54} T^{12} \) |
| 7 | \( 1 - 127514750 T^{2} + 154108736216303 p^{2} T^{4} - \)\(13\!\cdots\!00\)\( p^{4} T^{6} + 154108736216303 p^{20} T^{8} - 127514750 p^{36} T^{10} + p^{54} T^{12} \) |
| 11 | \( ( 1 - 54699 T + 5816187440 T^{2} - 182671409832855 T^{3} + 5816187440 p^{9} T^{4} - 54699 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 13 | \( 1 - 15844411230 T^{2} + \)\(30\!\cdots\!87\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(30\!\cdots\!87\)\( p^{18} T^{8} - 15844411230 p^{36} T^{10} + p^{54} T^{12} \) |
| 17 | \( 1 - 597847713835 T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(24\!\cdots\!55\)\( T^{6} + \)\(15\!\cdots\!02\)\( p^{18} T^{8} - 597847713835 p^{36} T^{10} + p^{54} T^{12} \) |
| 19 | \( ( 1 + 818845 T + 468173656712 T^{2} + 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} + 818845 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 23 | \( 1 - 215760622290 p T^{2} + \)\(14\!\cdots\!07\)\( T^{4} - \)\(32\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!07\)\( p^{18} T^{8} - 215760622290 p^{37} T^{10} + p^{54} T^{12} \) |
| 29 | \( ( 1 - 2175480 T + 1119231270883 p T^{2} - 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} - 2175480 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 31 | \( ( 1 - 4274066 T + 82851493809465 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 37 | \( 1 - 575490501959730 T^{2} + \)\(15\!\cdots\!87\)\( T^{4} - \)\(24\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!87\)\( p^{18} T^{8} - 575490501959730 p^{36} T^{10} + p^{54} T^{12} \) |
| 41 | \( ( 1 + 5926311 T + 232043727124790 T^{2} + \)\(23\!\cdots\!95\)\( T^{3} + 232043727124790 p^{9} T^{4} + 5926311 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 43 | \( 1 - 2755277785986450 T^{2} + \)\(32\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!00\)\( p^{2} T^{6} + \)\(32\!\cdots\!47\)\( p^{18} T^{8} - 2755277785986450 p^{36} T^{10} + p^{54} T^{12} \) |
| 47 | \( 1 - 4564445424701290 T^{2} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!67\)\( p^{18} T^{8} - 4564445424701290 p^{36} T^{10} + p^{54} T^{12} \) |
| 53 | \( 1 - 830296241726290 T^{2} + \)\(19\!\cdots\!67\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!67\)\( p^{18} T^{8} - 830296241726290 p^{36} T^{10} + p^{54} T^{12} \) |
| 59 | \( ( 1 - 5670960 T + 19337695182532817 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + 19337695182532817 p^{9} T^{4} - 5670960 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 61 | \( ( 1 - 125306926 T + 22192130203358915 T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 67 | \( 1 - 53234092789742135 T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!55\)\( T^{6} + \)\(12\!\cdots\!02\)\( p^{18} T^{8} - 53234092789742135 p^{36} T^{10} + p^{54} T^{12} \) |
| 71 | \( ( 1 + 297550596 T + 87036096018332165 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + 87036096018332165 p^{9} T^{4} + 297550596 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 73 | \( 1 - 155852935645686795 T^{2} + \)\(17\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!35\)\( T^{6} + \)\(17\!\cdots\!82\)\( p^{18} T^{8} - 155852935645686795 p^{36} T^{10} + p^{54} T^{12} \) |
| 79 | \( ( 1 - 310025170 T + 176092553119892457 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + 176092553119892457 p^{9} T^{4} - 310025170 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 83 | \( 1 - 836375861147057535 T^{2} + \)\(32\!\cdots\!02\)\( T^{4} - \)\(77\!\cdots\!55\)\( T^{6} + \)\(32\!\cdots\!02\)\( p^{18} T^{8} - 836375861147057535 p^{36} T^{10} + p^{54} T^{12} \) |
| 89 | \( ( 1 + 1103860035 T + 1320664213408319502 T^{2} + \)\(75\!\cdots\!55\)\( T^{3} + 1320664213408319502 p^{9} T^{4} + 1103860035 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 97 | \( 1 - 2865108733311184890 T^{2} + \)\(43\!\cdots\!67\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!67\)\( p^{18} T^{8} - 2865108733311184890 p^{36} T^{10} + p^{54} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.17880045452176393403046870667, −4.96247274055776040214507280723, −4.49059109712202586652093410781, −4.47627331944418533252162300249, −4.09870155706071635264801077706, −4.06054840265851292290251071992, −4.01479829323062086452700255149, −3.78584776682990557627386457104, −3.62321642007680720238077254826, −3.28949724990134381274324889217, −2.76663106215943924718595896436, −2.70665376275146823749822563895, −2.55095827534036877860324635697, −2.53270981785841252636491931489, −2.46376861918762420813662163323, −2.20513588346844288520838154655, −1.66626644936066422820405993207, −1.65279819035062547416512195426, −1.55871152507506019415299634536, −1.32481699074159740921739054709, −1.10031449984650016470891852110, −0.925939221564046206313172331247, −0.55492680600481305143433476768, −0.34958379675838115518790244956, −0.10635274859940219100692741282,
0.10635274859940219100692741282, 0.34958379675838115518790244956, 0.55492680600481305143433476768, 0.925939221564046206313172331247, 1.10031449984650016470891852110, 1.32481699074159740921739054709, 1.55871152507506019415299634536, 1.65279819035062547416512195426, 1.66626644936066422820405993207, 2.20513588346844288520838154655, 2.46376861918762420813662163323, 2.53270981785841252636491931489, 2.55095827534036877860324635697, 2.70665376275146823749822563895, 2.76663106215943924718595896436, 3.28949724990134381274324889217, 3.62321642007680720238077254826, 3.78584776682990557627386457104, 4.01479829323062086452700255149, 4.06054840265851292290251071992, 4.09870155706071635264801077706, 4.47627331944418533252162300249, 4.49059109712202586652093410781, 4.96247274055776040214507280723, 5.17880045452176393403046870667
Plot not available for L-functions of degree greater than 10.
