L(s) = 1 | − 5-s − 2·7-s + 9-s − 5·11-s − 8·17-s − 6·19-s − 23-s + 2·35-s + 2·43-s − 45-s + 2·47-s + 49-s + 5·55-s + 2·61-s − 2·63-s + 2·73-s + 10·77-s + 81-s + 2·83-s + 8·85-s + 6·95-s − 5·99-s − 8·101-s + 115-s + 16·119-s + 10·121-s − 125-s + ⋯ |
L(s) = 1 | − 5-s − 2·7-s + 9-s − 5·11-s − 8·17-s − 6·19-s − 23-s + 2·35-s + 2·43-s − 45-s + 2·47-s + 49-s + 5·55-s + 2·61-s − 2·63-s + 2·73-s + 10·77-s + 81-s + 2·83-s + 8·85-s + 6·95-s − 5·99-s − 8·101-s + 115-s + 16·119-s + 10·121-s − 125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03594474745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03594474745\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 19 | \( ( 1 + T + T^{2} )^{6} \) |
good | 3 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 11 | \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 17 | \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 43 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 97 | \( ( 1 - T )^{12}( 1 + T )^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.71080239285978576055351894160, −2.65899609260268344223543869833, −2.53590723157049831062490571791, −2.52046774174746375665532139051, −2.42406841697838773669788514475, −2.35737654000609618325822366356, −2.34199442303453918256386676525, −2.21346090922106945398782406575, −2.17805491461894684053077991308, −2.15638512218014445912180281272, −2.10601854173799218251851323213, −2.02600599758455615190836258333, −1.94945831837895790343847654678, −1.84107507055766738721045278617, −1.56465631318067239855286659987, −1.45863356054543155885098475401, −1.43801208066120929083031849009, −1.36787123244115362991737371481, −1.10742547221934459963893636076, −1.09835653041142875388264226897, −0.66259167544785012370933669529, −0.42292451016538524003399470322, −0.39244817662955645289995198493, −0.27356709359469144215434087550, −0.22656467052416317549442946369,
0.22656467052416317549442946369, 0.27356709359469144215434087550, 0.39244817662955645289995198493, 0.42292451016538524003399470322, 0.66259167544785012370933669529, 1.09835653041142875388264226897, 1.10742547221934459963893636076, 1.36787123244115362991737371481, 1.43801208066120929083031849009, 1.45863356054543155885098475401, 1.56465631318067239855286659987, 1.84107507055766738721045278617, 1.94945831837895790343847654678, 2.02600599758455615190836258333, 2.10601854173799218251851323213, 2.15638512218014445912180281272, 2.17805491461894684053077991308, 2.21346090922106945398782406575, 2.34199442303453918256386676525, 2.35737654000609618325822366356, 2.42406841697838773669788514475, 2.52046774174746375665532139051, 2.53590723157049831062490571791, 2.65899609260268344223543869833, 2.71080239285978576055351894160
Plot not available for L-functions of degree greater than 10.