| L(s) = 1 | + 815·9-s + 1.09e5·11-s + 1.63e6·19-s − 4.35e6·29-s − 8.54e6·31-s + 1.18e7·41-s + 1.27e8·49-s + 1.13e7·59-s + 2.50e8·61-s − 5.95e8·71-s − 6.20e8·79-s + 1.26e8·81-s + 2.20e9·89-s + 8.91e7·99-s − 9.16e8·101-s − 4.25e9·109-s − 2.65e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.58e10·169-s + ⋯ |
| L(s) = 1 | + 0.0414·9-s + 2.25·11-s + 2.88·19-s − 1.14·29-s − 1.66·31-s + 0.655·41-s + 3.15·49-s + 0.121·59-s + 2.31·61-s − 2.77·71-s − 1.79·79-s + 0.326·81-s + 3.72·89-s + 0.0932·99-s − 0.876·101-s − 2.88·109-s − 1.12·121-s + 1.49·169-s + 0.119·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5418057676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5418057676\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 815 T^{2} - 13997962 p^{2} T^{4} - 82628689595 p^{4} T^{6} - 13997962 p^{20} T^{8} - 815 p^{36} T^{10} + p^{54} T^{12} \) |
| 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 | \( 1 - 2865108733311184890 T^{2} + \)\(43\!\cdots\!67\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!67\)\( p^{18} T^{8} - 2865108733311184890 p^{36} T^{10} + p^{54} T^{12} \) |
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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
−4.68173064625099262439578799867, −4.12284884774896575050444350530, −4.07125541120097674689178787782, −3.94310635609647069876504552275, −3.93454477956409136689438397863, −3.90241198539099715163165138224, −3.59781302693165884322551604036, −3.44441584508934158966610573332, −3.12461807212778639902600656244, −2.80665208883509954974571775726, −2.79742379647505793393120930995, −2.78772903265664482666482944416, −2.54902659804746315498872973674, −2.04894280587728252100971859916, −2.03464283437350016911880813373, −1.70898977459799241955343976111, −1.61621447037682456041063898463, −1.46571376687663524340359609382, −1.42222287757630937338537115756, −0.857807587561656437438427072657, −0.835762965368064448110009432000, −0.810679349746540469182564517988, −0.75960086684049779893969625301, −0.32263862312116705775903853131, −0.03369591103947671414233440183,
0.03369591103947671414233440183, 0.32263862312116705775903853131, 0.75960086684049779893969625301, 0.810679349746540469182564517988, 0.835762965368064448110009432000, 0.857807587561656437438427072657, 1.42222287757630937338537115756, 1.46571376687663524340359609382, 1.61621447037682456041063898463, 1.70898977459799241955343976111, 2.03464283437350016911880813373, 2.04894280587728252100971859916, 2.54902659804746315498872973674, 2.78772903265664482666482944416, 2.79742379647505793393120930995, 2.80665208883509954974571775726, 3.12461807212778639902600656244, 3.44441584508934158966610573332, 3.59781302693165884322551604036, 3.90241198539099715163165138224, 3.93454477956409136689438397863, 3.94310635609647069876504552275, 4.07125541120097674689178787782, 4.12284884774896575050444350530, 4.68173064625099262439578799867
Plot not available for L-functions of degree greater than 10.
