| L(s) = 1 | + 23·4-s − 4·7-s − 566·13-s − 95·16-s + 1.12e3·19-s − 92·28-s − 2.58e3·31-s − 2.04e3·37-s − 1.92e3·43-s − 3.80e3·49-s − 1.30e4·52-s − 1.16e4·61-s − 4.65e3·64-s − 1.28e3·67-s + 2.58e4·73-s + 2.58e4·76-s + 2.56e4·79-s + 2.26e3·91-s + 2.77e4·97-s + 4.18e4·103-s + 4.41e4·109-s + 380·112-s + 1.40e3·121-s − 5.93e4·124-s + 127-s + 131-s − 4.49e3·133-s + ⋯ |
| L(s) = 1 | + 1.43·4-s − 0.0816·7-s − 3.34·13-s − 0.371·16-s + 3.11·19-s − 0.117·28-s − 2.68·31-s − 1.49·37-s − 1.03·43-s − 1.58·49-s − 4.81·52-s − 3.13·61-s − 1.13·64-s − 0.286·67-s + 4.84·73-s + 4.47·76-s + 4.11·79-s + 0.273·91-s + 2.95·97-s + 3.94·103-s + 3.71·109-s + 0.0302·112-s + 0.0962·121-s − 3.86·124-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.254·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.673584253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673584253\) |
| \(L(3)\) |
|
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 | \( 1 - 23 T^{2} + 39 p^{4} T^{4} - 2971 p^{2} T^{6} + 39 p^{12} T^{8} - 23 p^{16} T^{10} + p^{24} T^{12} \) |
| 7 | \( ( 1 + 2 T + 1908 T^{2} + 143488 T^{3} + 1908 p^{4} T^{4} + 2 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 11 | \( 1 - 1409 T^{2} + 58106163 T^{4} - 5681685529750 T^{6} + 58106163 p^{8} T^{8} - 1409 p^{16} T^{10} + p^{24} T^{12} \) |
| 13 | \( ( 1 + 283 T + 51998 T^{2} + 6631591 T^{3} + 51998 p^{4} T^{4} + 283 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 17 | \( 1 - 17497 p T^{2} + 46203670923 T^{4} - 4601250314251270 T^{6} + 46203670923 p^{8} T^{8} - 17497 p^{17} T^{10} + p^{24} T^{12} \) |
| 19 | \( ( 1 - 562 T + 480716 T^{2} - 149979988 T^{3} + 480716 p^{4} T^{4} - 562 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 23 | \( 1 - 974049 T^{2} + 459299732883 T^{4} - 147401128285900310 T^{6} + 459299732883 p^{8} T^{8} - 974049 p^{16} T^{10} + p^{24} T^{12} \) |
| 29 | \( 1 - 1385033 T^{2} + 1225196535459 T^{4} - 744677995536416614 T^{6} + 1225196535459 p^{8} T^{8} - 1385033 p^{16} T^{10} + p^{24} T^{12} \) |
| 31 | \( ( 1 + 1291 T + 1445403 T^{2} + 867982802 T^{3} + 1445403 p^{4} T^{4} + 1291 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 37 | \( ( 1 + 1020 T + 2642736 T^{2} + 1508413394 T^{3} + 2642736 p^{4} T^{4} + 1020 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 41 | \( 1 - 380554 p T^{2} + 104810670636303 T^{4} - \)\(38\!\cdots\!40\)\( T^{6} + 104810670636303 p^{8} T^{8} - 380554 p^{17} T^{10} + p^{24} T^{12} \) |
| 43 | \( ( 1 + 961 T + 10002323 T^{2} + 6263164342 T^{3} + 10002323 p^{4} T^{4} + 961 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 47 | \( 1 - 9331553 T^{2} + 13882498904139 T^{4} + 76581256018749525866 T^{6} + 13882498904139 p^{8} T^{8} - 9331553 p^{16} T^{10} + p^{24} T^{12} \) |
| 53 | \( 1 - 39800858 T^{2} + 709940849307759 T^{4} - \)\(72\!\cdots\!64\)\( T^{6} + 709940849307759 p^{8} T^{8} - 39800858 p^{16} T^{10} + p^{24} T^{12} \) |
| 59 | \( 1 + 2554174 T^{2} + 279886014468255 T^{4} + \)\(88\!\cdots\!20\)\( T^{6} + 279886014468255 p^{8} T^{8} + 2554174 p^{16} T^{10} + p^{24} T^{12} \) |
| 61 | \( ( 1 + 5828 T + 28797176 T^{2} + 86562776522 T^{3} + 28797176 p^{4} T^{4} + 5828 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 67 | \( ( 1 + 644 T + 42538080 T^{2} + 2808404230 T^{3} + 42538080 p^{4} T^{4} + 644 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 71 | \( 1 - 133602638 T^{2} + 7846441935466239 T^{4} - \)\(25\!\cdots\!64\)\( T^{6} + 7846441935466239 p^{8} T^{8} - 133602638 p^{16} T^{10} + p^{24} T^{12} \) |
| 73 | \( ( 1 - 12914 T + 103957388 T^{2} - 592942929968 T^{3} + 103957388 p^{4} T^{4} - 12914 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 79 | \( ( 1 - 12845 T + 122866866 T^{2} - 908074925101 T^{3} + 122866866 p^{4} T^{4} - 12845 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 83 | \( 1 + 36874182 T^{2} + 5308384605336879 T^{4} + \)\(15\!\cdots\!36\)\( T^{6} + 5308384605336879 p^{8} T^{8} + 36874182 p^{16} T^{10} + p^{24} T^{12} \) |
| 89 | \( 1 - 282046614 T^{2} + 37067524318715343 T^{4} - \)\(29\!\cdots\!00\)\( T^{6} + 37067524318715343 p^{8} T^{8} - 282046614 p^{16} T^{10} + p^{24} T^{12} \) |
| 97 | \( ( 1 - 13896 T + 289086552 T^{2} - 2385107095066 T^{3} + 289086552 p^{4} T^{4} - 13896 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
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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
−5.08696845333122977176095991820, −4.82762683923588113458041725278, −4.77777847881113828477107017193, −4.57129107515284096556254611261, −4.51013082962634257344084806266, −3.96087242921100081806786100584, −3.84989212375203885441631738984, −3.55375294851042435312283671180, −3.51061604803363698133392134927, −3.28613927474992742389954269449, −3.14155930058227580762213542122, −3.03948926447561094191050758606, −2.96105952204788945513262535423, −2.44880046455041222637904813799, −2.13180075857946044865139482010, −2.11661961535101083355442566974, −2.08647123221471822868061739479, −1.97639365890846256424843585003, −1.83370810624388834479726124093, −1.30283850177815215643188488478, −1.12714986224364502634455440880, −0.789069643843222979205079957521, −0.61517252379642877138922963805, −0.35370877925415895731381481949, −0.10929154521977961813826437722,
0.10929154521977961813826437722, 0.35370877925415895731381481949, 0.61517252379642877138922963805, 0.789069643843222979205079957521, 1.12714986224364502634455440880, 1.30283850177815215643188488478, 1.83370810624388834479726124093, 1.97639365890846256424843585003, 2.08647123221471822868061739479, 2.11661961535101083355442566974, 2.13180075857946044865139482010, 2.44880046455041222637904813799, 2.96105952204788945513262535423, 3.03948926447561094191050758606, 3.14155930058227580762213542122, 3.28613927474992742389954269449, 3.51061604803363698133392134927, 3.55375294851042435312283671180, 3.84989212375203885441631738984, 3.96087242921100081806786100584, 4.51013082962634257344084806266, 4.57129107515284096556254611261, 4.77777847881113828477107017193, 4.82762683923588113458041725278, 5.08696845333122977176095991820
Plot not available for L-functions of degree greater than 10.
