| L(s) = 1 | − 6·3-s + 3·4-s + 15·9-s − 18·12-s + 3·16-s + 24·17-s + 32·23-s + 26·25-s − 14·27-s − 26·29-s + 45·36-s − 16·43-s − 18·48-s − 25·49-s − 144·51-s + 60·53-s + 20·61-s − 2·64-s + 72·68-s − 192·69-s − 156·75-s − 20·79-s − 21·81-s + 156·87-s + 96·92-s + 78·100-s + 10·101-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 3/2·4-s + 5·9-s − 5.19·12-s + 3/4·16-s + 5.82·17-s + 6.67·23-s + 26/5·25-s − 2.69·27-s − 4.82·29-s + 15/2·36-s − 2.43·43-s − 2.59·48-s − 3.57·49-s − 20.1·51-s + 8.24·53-s + 2.56·61-s − 1/4·64-s + 8.73·68-s − 23.1·69-s − 18.0·75-s − 2.25·79-s − 7/3·81-s + 16.7·87-s + 10.0·92-s + 39/5·100-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.04767810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.04767810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 3 | \( ( 1 + T + T^{2} )^{6} \) |
| 13 | \( 1 \) |
| good | 5 | \( ( 1 - 13 T^{2} + 9 p T^{4} - 49 T^{6} + 9 p^{3} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 + 25 T^{2} + 356 T^{4} + 389 p T^{6} + 9155 T^{8} - 60756 T^{10} - 807015 T^{12} - 60756 p^{2} T^{14} + 9155 p^{4} T^{16} + 389 p^{7} T^{18} + 356 p^{8} T^{20} + 25 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 + 3 p T^{2} + 496 T^{4} + 4463 T^{6} + 30647 T^{8} + 226384 T^{10} + 2097353 T^{12} + 226384 p^{2} T^{14} + 30647 p^{4} T^{16} + 4463 p^{6} T^{18} + 496 p^{8} T^{20} + 3 p^{11} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 - 12 T + 73 T^{2} - 244 T^{3} + 186 T^{4} + 3916 T^{5} - 25071 T^{6} + 3916 p T^{7} + 186 p^{2} T^{8} - 244 p^{3} T^{9} + 73 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 19 | \( 1 + 34 T^{2} + 15 p T^{4} - 6730 T^{6} - 173638 T^{8} - 1122582 T^{10} - 207131 T^{12} - 1122582 p^{2} T^{14} - 173638 p^{4} T^{16} - 6730 p^{6} T^{18} + 15 p^{9} T^{20} + 34 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( ( 1 - 16 T + 111 T^{2} - 640 T^{3} + 4250 T^{4} - 23544 T^{5} + 110935 T^{6} - 23544 p T^{7} + 4250 p^{2} T^{8} - 640 p^{3} T^{9} + 111 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 13 T + 70 T^{2} + 225 T^{3} + 27 T^{4} - 11144 T^{5} - 95619 T^{6} - 11144 p T^{7} + 27 p^{2} T^{8} + 225 p^{3} T^{9} + 70 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 61 T^{2} - 79 T^{4} + 63399 T^{6} - 79 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( 1 + 118 T^{2} + 8573 T^{4} + 283154 T^{6} + 760106 T^{8} - 575891154 T^{10} - 29841263691 T^{12} - 575891154 p^{2} T^{14} + 760106 p^{4} T^{16} + 283154 p^{6} T^{18} + 8573 p^{8} T^{20} + 118 p^{10} T^{22} + p^{12} T^{24} \) |
| 41 | \( 1 + 162 T^{2} + 13237 T^{4} + 793862 T^{6} + 40942250 T^{8} + 1914453226 T^{10} + 82116748925 T^{12} + 1914453226 p^{2} T^{14} + 40942250 p^{4} T^{16} + 793862 p^{6} T^{18} + 13237 p^{8} T^{20} + 162 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 8 T - 21 T^{2} - 8 T^{3} + 818 T^{4} - 336 p T^{5} - 3527 p T^{6} - 336 p^{2} T^{7} + 818 p^{2} T^{8} - 8 p^{3} T^{9} - 21 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 202 T^{2} + 19631 T^{4} - 1156428 T^{6} + 19631 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 15 T + 87 T^{2} - 343 T^{3} + 87 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( 1 + 313 T^{2} + 55264 T^{4} + 6825319 T^{6} + 651340271 T^{8} + 50408102728 T^{10} + 3246178236377 T^{12} + 50408102728 p^{2} T^{14} + 651340271 p^{4} T^{16} + 6825319 p^{6} T^{18} + 55264 p^{8} T^{20} + 313 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 10 T - 107 T^{2} + 386 T^{3} + 17942 T^{4} - 38866 T^{5} - 1023943 T^{6} - 38866 p T^{7} + 17942 p^{2} T^{8} + 386 p^{3} T^{9} - 107 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 - 2 T^{2} - 6259 T^{4} - 456886 T^{6} + 11554922 T^{8} + 1463669526 T^{10} + 54118328613 T^{12} + 1463669526 p^{2} T^{14} + 11554922 p^{4} T^{16} - 456886 p^{6} T^{18} - 6259 p^{8} T^{20} - 2 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( 1 + 246 T^{2} + 29589 T^{4} + 2375106 T^{6} + 2210214 p T^{8} + 10466499918 T^{10} + 742679777981 T^{12} + 10466499918 p^{2} T^{14} + 2210214 p^{5} T^{16} + 2375106 p^{6} T^{18} + 29589 p^{8} T^{20} + 246 p^{10} T^{22} + p^{12} T^{24} \) |
| 73 | \( ( 1 - 369 T^{2} + 60401 T^{4} - 5663609 T^{6} + 60401 p^{2} T^{8} - 369 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 5 T + 33 T^{2} - 679 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 - T^{2} + 9185 T^{4} - 472977 T^{6} + 9185 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( 1 + 482 T^{2} + 131605 T^{4} + 25142534 T^{6} + 3706151978 T^{8} + 439782134954 T^{10} + 43001384383901 T^{12} + 439782134954 p^{2} T^{14} + 3706151978 p^{4} T^{16} + 25142534 p^{6} T^{18} + 131605 p^{8} T^{20} + 482 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( 1 + 505 T^{2} + 143472 T^{4} + 28407287 T^{6} + 4342907687 T^{8} + 542278176552 T^{10} + 56981517525073 T^{12} + 542278176552 p^{2} T^{14} + 4342907687 p^{4} T^{16} + 28407287 p^{6} T^{18} + 143472 p^{8} T^{20} + 505 p^{10} T^{22} + p^{12} T^{24} \) |
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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
−3.23413934408203532929953114074, −3.18416459025987822012642301842, −3.03912022321123794418160505883, −2.93341651136908020287216057062, −2.84001474178045465692622315449, −2.71050148869356301557517713413, −2.65673302535476133473879056804, −2.51169858205239907591643534146, −2.38646054006791625134738681855, −2.28464771616379812472662900657, −2.27931285492760239260437404053, −1.99637027455411371084570881718, −1.80665703023923259608772885903, −1.73590671613065398927896636591, −1.64917370349581000121612116177, −1.34523917737577544082594581245, −1.23765240517610202480901393710, −1.21241266981680682007122827477, −1.18188503948486696901808433231, −1.10294719047501977760319502441, −0.855631390080655018313249176957, −0.77969031093146185612092570878, −0.65402525548346316310225693858, −0.53244452383544422801633605521, −0.28378706893365250370537829771,
0.28378706893365250370537829771, 0.53244452383544422801633605521, 0.65402525548346316310225693858, 0.77969031093146185612092570878, 0.855631390080655018313249176957, 1.10294719047501977760319502441, 1.18188503948486696901808433231, 1.21241266981680682007122827477, 1.23765240517610202480901393710, 1.34523917737577544082594581245, 1.64917370349581000121612116177, 1.73590671613065398927896636591, 1.80665703023923259608772885903, 1.99637027455411371084570881718, 2.27931285492760239260437404053, 2.28464771616379812472662900657, 2.38646054006791625134738681855, 2.51169858205239907591643534146, 2.65673302535476133473879056804, 2.71050148869356301557517713413, 2.84001474178045465692622315449, 2.93341651136908020287216057062, 3.03912022321123794418160505883, 3.18416459025987822012642301842, 3.23413934408203532929953114074
Plot not available for L-functions of degree greater than 10.
