L(s) = 1 | − 6·3-s − 5·4-s + 21·9-s + 30·12-s + 7·16-s + 2·17-s + 4·25-s − 56·27-s − 4·29-s − 105·36-s + 30·43-s − 42·48-s + 36·49-s − 12·51-s − 34·53-s − 26·61-s + 14·64-s − 10·68-s − 24·75-s + 6·79-s + 126·81-s + 24·87-s − 20·100-s − 4·101-s + 28·103-s − 26·107-s + 280·108-s + ⋯ |
L(s) = 1 | − 3.46·3-s − 5/2·4-s + 7·9-s + 8.66·12-s + 7/4·16-s + 0.485·17-s + 4/5·25-s − 10.7·27-s − 0.742·29-s − 17.5·36-s + 4.57·43-s − 6.06·48-s + 36/7·49-s − 1.68·51-s − 4.67·53-s − 3.32·61-s + 7/4·64-s − 1.21·68-s − 2.77·75-s + 0.675·79-s + 14·81-s + 2.57·87-s − 2·100-s − 0.398·101-s + 2.75·103-s − 2.51·107-s + 26.9·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3656040529\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3656040529\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T )^{6} \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 5 T^{2} + 9 p T^{4} + 41 T^{6} + 9 p^{3} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 - 4 T^{2} + 36 T^{4} - 241 T^{6} + 36 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 36 T^{2} + 572 T^{4} - 5165 T^{6} + 572 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 25 T^{2} + 485 T^{4} - 5601 T^{6} + 485 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - T + 35 T^{2} - 47 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 93 T^{2} + 3917 T^{4} - 95369 T^{6} + 3917 p^{2} T^{8} - 93 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 12 T^{2} + 824 T^{4} - 48797 T^{6} + 824 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 56 T^{2} + 4660 T^{4} - 150689 T^{6} + 4660 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 173 T^{2} + 14457 T^{4} - 740849 T^{6} + 14457 p^{2} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 15 T + 176 T^{2} - 1331 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 261 T^{2} + 29285 T^{4} - 1807289 T^{6} + 29285 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 286 T^{2} + 37671 T^{4} - 2853988 T^{6} + 37671 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 189 T^{2} + 11465 T^{4} - 439313 T^{6} + 11465 p^{2} T^{8} - 189 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 244 T^{2} + 32208 T^{4} - 2788141 T^{6} + 32208 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 132 T^{2} + 19884 T^{4} - 1422313 T^{6} + 19884 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 436 T^{2} + 83400 T^{4} - 8978917 T^{6} + 83400 p^{2} T^{8} - 436 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 333 T^{2} + 57485 T^{4} - 6354113 T^{6} + 57485 p^{2} T^{8} - 333 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 5 T^{2} + 8710 T^{4} - 929407 T^{6} + 8710 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.78404864299300655338576803600, −5.76165902909497038600322071581, −5.68280000537407585093511917077, −5.32653231624079750034862028456, −5.09253788230750429277287069831, −4.99029319833712095605959702493, −4.77739702292958854066591356164, −4.76981986409579124471295970560, −4.48571502182304183484283474871, −4.33565236143602620048484189321, −4.29352358505292305525147263723, −4.17382617570952520916689324090, −3.78191431444966897064624159984, −3.74934270031281995409354816353, −3.39744102875102796300558873916, −3.23944554141581949554549581463, −2.65229395028605778009515408795, −2.62928793339349890588837605010, −2.35604807238106439564765285765, −1.81395735862118071612012367370, −1.54336874929368138992200971270, −1.35479947325132124480639002243, −0.61856969494096997661188266923, −0.57744192035166526927365774838, −0.54673112717218748091963727800,
0.54673112717218748091963727800, 0.57744192035166526927365774838, 0.61856969494096997661188266923, 1.35479947325132124480639002243, 1.54336874929368138992200971270, 1.81395735862118071612012367370, 2.35604807238106439564765285765, 2.62928793339349890588837605010, 2.65229395028605778009515408795, 3.23944554141581949554549581463, 3.39744102875102796300558873916, 3.74934270031281995409354816353, 3.78191431444966897064624159984, 4.17382617570952520916689324090, 4.29352358505292305525147263723, 4.33565236143602620048484189321, 4.48571502182304183484283474871, 4.76981986409579124471295970560, 4.77739702292958854066591356164, 4.99029319833712095605959702493, 5.09253788230750429277287069831, 5.32653231624079750034862028456, 5.68280000537407585093511917077, 5.76165902909497038600322071581, 5.78404864299300655338576803600
Plot not available for L-functions of degree greater than 10.