| L(s) = 1 | − 4-s + 2·7-s + 16-s + 25-s − 2·28-s − 6·31-s + 3·49-s + 2·79-s − 100-s + 2·112-s + 121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s + 16-s + 25-s − 2·28-s − 6·31-s + 3·49-s + 2·79-s − 100-s + 2·112-s + 121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.331236021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331236021\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| good | 5 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 13 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{12} \) |
| 23 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 29 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 61 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{12}( 1 + T + T^{2} )^{12} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 83 | \( ( 1 - T^{2} + T^{4} )^{6}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 97 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.75864809981307029772636172584, −1.73519341401985450997068862956, −1.70873067938099022490237462227, −1.67739150551778572664969231956, −1.64850857802746986346380436302, −1.56680934053516157428729409084, −1.52660195330248262100242620880, −1.47534586839237768646352364996, −1.37953422075523856297589078610, −1.29315290256264806755649089514, −1.26975304375554556739999878772, −1.22550376837776529102033516439, −1.09505015546404126750648100756, −1.08361669391317668251335068883, −0.977391265255056606794434867000, −0.921186310074318889255220034029, −0.918925978638913004094422286160, −0.825228992129413976653640403835, −0.76190288290450746138197225704, −0.75454546426788277584859318521, −0.73827356177538074995404995887, −0.55892190719950638605345557258, −0.43277658799497628424043042551, −0.42871042162992772998181664506, −0.13083142315937017604533364868,
0.13083142315937017604533364868, 0.42871042162992772998181664506, 0.43277658799497628424043042551, 0.55892190719950638605345557258, 0.73827356177538074995404995887, 0.75454546426788277584859318521, 0.76190288290450746138197225704, 0.825228992129413976653640403835, 0.918925978638913004094422286160, 0.921186310074318889255220034029, 0.977391265255056606794434867000, 1.08361669391317668251335068883, 1.09505015546404126750648100756, 1.22550376837776529102033516439, 1.26975304375554556739999878772, 1.29315290256264806755649089514, 1.37953422075523856297589078610, 1.47534586839237768646352364996, 1.52660195330248262100242620880, 1.56680934053516157428729409084, 1.64850857802746986346380436302, 1.67739150551778572664969231956, 1.70873067938099022490237462227, 1.73519341401985450997068862956, 1.75864809981307029772636172584
Plot not available for L-functions of degree greater than 10.
