| L(s) = 1 | − 3·4-s + 4·9-s + 4·11-s + 6·16-s − 6·19-s − 12·29-s + 16·31-s − 12·36-s − 12·44-s + 12·49-s − 12·59-s − 4·61-s − 10·64-s − 12·71-s + 18·76-s − 40·79-s + 12·81-s − 28·89-s + 16·99-s − 16·101-s − 4·109-s + 36·116-s − 6·121-s − 48·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 4/3·9-s + 1.20·11-s + 3/2·16-s − 1.37·19-s − 2.22·29-s + 2.87·31-s − 2·36-s − 1.80·44-s + 12/7·49-s − 1.56·59-s − 0.512·61-s − 5/4·64-s − 1.42·71-s + 2.06·76-s − 4.50·79-s + 4/3·81-s − 2.96·89-s + 1.60·99-s − 1.59·101-s − 0.383·109-s + 3.34·116-s − 0.545·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6383805949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6383805949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - 4 T^{2} + 4 T^{4} + p T^{6} + 4 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 12 T^{2} + 60 T^{4} - 349 T^{6} + 60 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 2 T + 9 T^{2} - 20 T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 36 T^{2} + 60 p T^{4} - 11281 T^{6} + 60 p^{3} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 24 T^{2} + 336 T^{4} - 7297 T^{6} + 336 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 32 T^{2} + 340 T^{4} + 6203 T^{6} + 340 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + 84 T^{2} + 339 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 8 T + 89 T^{2} - 440 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 171 T^{2} + 13614 T^{4} - 640543 T^{6} + 13614 p^{2} T^{8} - 171 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + p T^{2} )^{6} \) |
| 43 | \( 1 - 142 T^{2} + 10055 T^{4} - 498052 T^{6} + 10055 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 159 T^{2} + 12990 T^{4} - 724219 T^{6} + 12990 p^{2} T^{8} - 159 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 8 T^{2} + 4780 T^{4} - 7801 T^{6} + 4780 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 6 T - 24 T^{2} - 723 T^{3} - 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 2 T + 127 T^{2} + 124 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 340 T^{2} + 51848 T^{4} - 4491013 T^{6} + 51848 p^{2} T^{8} - 340 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 6 T + 21 T^{2} - 660 T^{3} + 21 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 240 T^{2} + 30456 T^{4} - 2566201 T^{6} + 30456 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 20 T + 269 T^{2} + 2840 T^{3} + 269 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 266 T^{2} + 36631 T^{4} - 3531148 T^{6} + 36631 p^{2} T^{8} - 266 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 14 T + 231 T^{2} + 2180 T^{3} + 231 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 114 T^{2} + 9663 T^{4} + 108380 T^{6} + 9663 p^{2} T^{8} + 114 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.46567926164392620434802941542, −5.18072649887683061693478661579, −4.96322753448470782834708735263, −4.73437425656294287503114842348, −4.44920278689761782626673028891, −4.28615626997913101189517988385, −4.23765820805346630761349210753, −4.23274293104942551334939936796, −4.19194195233440441532132139118, −4.13547085172331797934534427530, −3.70441951959950249883393506760, −3.57170476355457033795059513807, −3.14929335660521045419880484611, −3.06556509149052843203318753050, −2.93137557232701202077408354688, −2.81305144509178528479848249391, −2.58727923991417659482539010778, −2.03683891189127788840186396488, −2.01469847215699411061785591196, −1.59737522439635978392269332245, −1.58364110978100423151755258519, −1.37384753281162657188321274580, −0.936750481410795245148169208529, −0.72714030816066104783210256445, −0.14929890827440778306419053904,
0.14929890827440778306419053904, 0.72714030816066104783210256445, 0.936750481410795245148169208529, 1.37384753281162657188321274580, 1.58364110978100423151755258519, 1.59737522439635978392269332245, 2.01469847215699411061785591196, 2.03683891189127788840186396488, 2.58727923991417659482539010778, 2.81305144509178528479848249391, 2.93137557232701202077408354688, 3.06556509149052843203318753050, 3.14929335660521045419880484611, 3.57170476355457033795059513807, 3.70441951959950249883393506760, 4.13547085172331797934534427530, 4.19194195233440441532132139118, 4.23274293104942551334939936796, 4.23765820805346630761349210753, 4.28615626997913101189517988385, 4.44920278689761782626673028891, 4.73437425656294287503114842348, 4.96322753448470782834708735263, 5.18072649887683061693478661579, 5.46567926164392620434802941542
Plot not available for L-functions of degree greater than 10.
