L(s) = 1 | + 6·2-s + 15·4-s − 3·5-s + 18·8-s − 18·10-s − 6·11-s + 3·13-s + 3·16-s − 6·17-s + 3·19-s − 45·20-s − 36·22-s − 12·23-s + 15·25-s + 18·26-s + 9·27-s − 9·29-s − 6·31-s − 30·32-s − 36·34-s + 3·37-s + 18·38-s − 54·40-s + 3·43-s − 90·44-s − 72·46-s + 6·47-s + ⋯ |
L(s) = 1 | + 4.24·2-s + 15/2·4-s − 1.34·5-s + 6.36·8-s − 5.69·10-s − 1.80·11-s + 0.832·13-s + 3/4·16-s − 1.45·17-s + 0.688·19-s − 10.0·20-s − 7.67·22-s − 2.50·23-s + 3·25-s + 3.53·26-s + 1.73·27-s − 1.67·29-s − 1.07·31-s − 5.30·32-s − 6.17·34-s + 0.493·37-s + 2.91·38-s − 8.53·40-s + 0.457·43-s − 13.5·44-s − 10.6·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.925937984\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.925937984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p^{2} T^{3} + p^{3} T^{6} \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 6 p T^{5} - 371 T^{6} + 6 p^{2} T^{7} + 69 p^{2} T^{8} - 9 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 852 p T^{7} + 492 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 6060 p T^{7} + 420 p^{2} T^{8} + 54 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 1026 p T^{7} + 8322 p^{2} T^{8} - 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 3 T + 87 T^{2} - 333 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 + 6 T - 114 T^{2} - 378 T^{3} + 10716 T^{4} + 17304 T^{5} - 587549 T^{6} + 17304 p T^{7} + 10716 p^{2} T^{8} - 378 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 3 T + 105 T^{2} + 405 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 126648 p T^{7} + 8088 p^{2} T^{8} + 702 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 12 T - 60 T^{2} - 198 T^{3} + 7584 T^{4} - 70800 T^{5} - 1684181 T^{6} - 70800 p T^{7} + 7584 p^{2} T^{8} - 198 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65399834147712095373816727147, −5.62594004901320904889964360985, −5.52996336151396633458072319392, −5.34966852562544007477651273867, −5.34202822079052198010951961552, −4.89114962578282056372463793187, −4.85554584725551785747773197947, −4.80731246549708467716868329916, −4.52269838339875983186760250700, −4.24520980253784602059303601803, −4.12621263038470736962608168780, −4.12173679141026505828035303466, −4.10203968912438284950744404112, −3.68941037594969802968216591028, −3.51186819505681221662884877115, −3.38157216903191666872861020199, −3.09697582623663454091332160905, −2.95724197247819951691978530924, −2.61051943714259769912099645038, −2.52109541255825529462245258740, −2.26320601607568600418535310076, −1.68847135581890028304984931045, −1.66855144921805087931143060598, −0.882887523815293774766513605622, −0.24802698083798666870835540963,
0.24802698083798666870835540963, 0.882887523815293774766513605622, 1.66855144921805087931143060598, 1.68847135581890028304984931045, 2.26320601607568600418535310076, 2.52109541255825529462245258740, 2.61051943714259769912099645038, 2.95724197247819951691978530924, 3.09697582623663454091332160905, 3.38157216903191666872861020199, 3.51186819505681221662884877115, 3.68941037594969802968216591028, 4.10203968912438284950744404112, 4.12173679141026505828035303466, 4.12621263038470736962608168780, 4.24520980253784602059303601803, 4.52269838339875983186760250700, 4.80731246549708467716868329916, 4.85554584725551785747773197947, 4.89114962578282056372463793187, 5.34202822079052198010951961552, 5.34966852562544007477651273867, 5.52996336151396633458072319392, 5.62594004901320904889964360985, 5.65399834147712095373816727147
Plot not available for L-functions of degree greater than 10.