| L(s) = 1 | + 10·3-s − 156·7-s + 13·9-s + 156·13-s + 4.50e3·19-s − 1.56e3·21-s + 5.75e4·25-s − 1.27e4·27-s + 7.42e4·31-s + 1.71e5·37-s + 1.56e3·39-s + 2.91e5·43-s − 8.19e4·49-s + 4.50e4·57-s + 5.92e5·61-s − 2.02e3·63-s + 5.70e5·67-s + 1.11e6·73-s + 5.75e5·75-s + 1.05e6·79-s − 3.11e5·81-s − 2.43e4·91-s + 7.42e5·93-s − 7.98e5·97-s + 7.46e5·103-s − 2.45e6·109-s + 1.71e6·111-s + ⋯ |
| L(s) = 1 | + 0.370·3-s − 0.454·7-s + 0.0178·9-s + 0.0710·13-s + 0.656·19-s − 0.168·21-s + 3.68·25-s − 0.646·27-s + 2.49·31-s + 3.37·37-s + 0.0262·39-s + 3.66·43-s − 0.696·49-s + 0.242·57-s + 2.60·61-s − 0.00811·63-s + 1.89·67-s + 2.87·73-s + 1.36·75-s + 2.13·79-s − 0.586·81-s − 0.0322·91-s + 0.923·93-s − 0.874·97-s + 0.683·103-s − 1.89·109-s + 1.25·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s+3)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(12.62180973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.62180973\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 10 T + 29 p T^{2} + 148 p^{4} T^{3} + 29 p^{7} T^{4} - 10 p^{12} T^{5} + p^{18} T^{6} \) |
| good | 5 | \( 1 - 57558 T^{2} + 66611991 p^{2} T^{4} - 50459973108 p^{4} T^{6} + 66611991 p^{14} T^{8} - 57558 p^{24} T^{10} + p^{36} T^{12} \) |
| 7 | \( ( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 50079 p^{6} T^{4} + 78 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 11 | \( 1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} + 13030923525375 p^{12} T^{8} - 3337110 p^{24} T^{10} + p^{36} T^{12} \) |
| 13 | \( ( 1 - 6 p T + 8645655 T^{2} - 5741493604 T^{3} + 8645655 p^{6} T^{4} - 6 p^{13} T^{5} + p^{18} T^{6} )^{2} \) |
| 17 | \( 1 - 63219270 T^{2} + 2233832285753679 T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + 2233832285753679 p^{12} T^{8} - 63219270 p^{24} T^{10} + p^{36} T^{12} \) |
| 19 | \( ( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 64193463 p^{6} T^{4} - 2250 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 23 | \( 1 + 37480026 T^{2} + 40240367212774575 T^{4} + \)\(51\!\cdots\!12\)\( T^{6} + 40240367212774575 p^{12} T^{8} + 37480026 p^{24} T^{10} + p^{36} T^{12} \) |
| 29 | \( 1 - 1660966710 T^{2} + 1523344801209102879 T^{4} - \)\(10\!\cdots\!08\)\( T^{6} + 1523344801209102879 p^{12} T^{8} - 1660966710 p^{24} T^{10} + p^{36} T^{12} \) |
| 31 | \( ( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2847433071 p^{6} T^{4} - 37122 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 37 | \( ( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 8033556999 p^{6} T^{4} - 85566 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 41 | \( 1 - 7569104550 T^{2} + 1363608137405888943 T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + 1363608137405888943 p^{12} T^{8} - 7569104550 p^{24} T^{10} + p^{36} T^{12} \) |
| 43 | \( ( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} + 21837177927 p^{6} T^{4} - 145530 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 47 | \( 1 - 52339293510 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!23\)\( p^{12} T^{8} - 52339293510 p^{24} T^{10} + p^{36} T^{12} \) |
| 53 | \( 1 - 75107476374 T^{2} + \)\(29\!\cdots\!67\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(29\!\cdots\!67\)\( p^{12} T^{8} - 75107476374 p^{24} T^{10} + p^{36} T^{12} \) |
| 59 | \( 1 - 180693503190 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(77\!\cdots\!36\)\( T^{6} + \)\(15\!\cdots\!31\)\( p^{12} T^{8} - 180693503190 p^{24} T^{10} + p^{36} T^{12} \) |
| 61 | \( ( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} + 42371190135 p^{6} T^{4} - 296046 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 67 | \( ( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} + 220735162839 p^{6} T^{4} - 285450 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 71 | \( 1 - 402983850726 T^{2} + \)\(77\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(77\!\cdots\!75\)\( p^{12} T^{8} - 402983850726 p^{24} T^{10} + p^{36} T^{12} \) |
| 73 | \( ( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} + 328149232287 p^{6} T^{4} - 559830 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 79 | \( ( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} + 705484689807 p^{6} T^{4} - 526818 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 83 | \( 1 - 1779363154038 T^{2} + \)\(13\!\cdots\!67\)\( T^{4} - \)\(58\!\cdots\!36\)\( T^{6} + \)\(13\!\cdots\!67\)\( p^{12} T^{8} - 1779363154038 p^{24} T^{10} + p^{36} T^{12} \) |
| 89 | \( 1 - 1464277161318 T^{2} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(68\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!39\)\( p^{12} T^{8} - 1464277161318 p^{24} T^{10} + p^{36} T^{12} \) |
| 97 | \( ( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} + 1108436400207 p^{6} T^{4} + 399258 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 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
−7.72655488189807329288142256754, −7.47431822012445070782612795423, −7.35224627573673099693310376564, −6.65303084230754778779926859427, −6.57589864664399616294330889710, −6.50237567677033793329494409072, −6.40164570611060507035531503177, −6.06607789808751555199274179005, −5.44530452595381994521676310946, −5.28159273344982012771145042059, −5.23703564821750490429340691164, −4.85714461043540613435511158323, −4.38646535495858543981012351283, −4.24247359918315630302175442444, −3.99465763948517162967677950245, −3.61576263944256281166057072559, −3.19557504539974704825985499660, −2.81505493946119503805598678888, −2.54420280282852575451194580206, −2.52951615452794427206754645794, −2.09343164787456057309870499590, −1.14340361775573487873736999239, −0.893452686496486292142571378422, −0.878710824804144684127494159821, −0.55472109528060079629770523674,
0.55472109528060079629770523674, 0.878710824804144684127494159821, 0.893452686496486292142571378422, 1.14340361775573487873736999239, 2.09343164787456057309870499590, 2.52951615452794427206754645794, 2.54420280282852575451194580206, 2.81505493946119503805598678888, 3.19557504539974704825985499660, 3.61576263944256281166057072559, 3.99465763948517162967677950245, 4.24247359918315630302175442444, 4.38646535495858543981012351283, 4.85714461043540613435511158323, 5.23703564821750490429340691164, 5.28159273344982012771145042059, 5.44530452595381994521676310946, 6.06607789808751555199274179005, 6.40164570611060507035531503177, 6.50237567677033793329494409072, 6.57589864664399616294330889710, 6.65303084230754778779926859427, 7.35224627573673099693310376564, 7.47431822012445070782612795423, 7.72655488189807329288142256754
Plot not available for L-functions of degree greater than 10.
