L(s) = 1 | + 8·5-s − 12·7-s + 18·9-s + 8·11-s − 24·19-s + 68·23-s + 37·25-s − 96·35-s − 208·37-s + 68·41-s + 144·45-s + 268·47-s − 106·49-s + 64·53-s + 64·55-s + 360·59-s − 216·63-s − 96·77-s + 181·81-s − 76·89-s − 192·95-s + 144·99-s − 124·103-s + 544·115-s − 134·121-s + 160·125-s + 127-s + ⋯ |
L(s) = 1 | + 8/5·5-s − 1.71·7-s + 2·9-s + 8/11·11-s − 1.26·19-s + 2.95·23-s + 1.47·25-s − 2.74·35-s − 5.62·37-s + 1.65·41-s + 16/5·45-s + 5.70·47-s − 2.16·49-s + 1.20·53-s + 1.16·55-s + 6.10·59-s − 3.42·63-s − 1.24·77-s + 2.23·81-s − 0.853·89-s − 2.02·95-s + 1.45·99-s − 1.20·103-s + 4.73·115-s − 1.10·121-s + 1.27·125-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(22.66971129\) |
\(L(\frac12)\) |
\(\approx\) |
\(22.66971129\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 8 T + 27 T^{2} - 16 p T^{3} + 27 p^{2} T^{4} - 8 p^{4} T^{5} + p^{6} T^{6} \) |
good | 3 | \( 1 - 2 p^{2} T^{2} + 143 T^{4} - 1052 T^{6} + 143 p^{4} T^{8} - 2 p^{10} T^{10} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 6 T + 107 T^{2} + 596 T^{3} + 107 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 - 4 T + 91 T^{2} + 632 T^{3} + 91 p^{2} T^{4} - 4 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( ( 1 + 23 p T^{2} + 64 p T^{3} + 23 p^{3} T^{4} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 38 p T^{2} + 265423 T^{4} - 87226900 T^{6} + 265423 p^{4} T^{8} - 38 p^{9} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 + 12 T + 299 T^{2} + 12056 T^{3} + 299 p^{2} T^{4} + 12 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( ( 1 - 34 T + 1435 T^{2} - 26812 T^{3} + 1435 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 29 | \( 1 - 2666 T^{2} + 3408319 T^{4} - 3146722508 T^{6} + 3408319 p^{4} T^{8} - 2666 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( 1 - 3206 T^{2} + 5912719 T^{4} - 6798206228 T^{6} + 5912719 p^{4} T^{8} - 3206 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( ( 1 + 104 T + 6811 T^{2} + 305552 T^{3} + 6811 p^{2} T^{4} + 104 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( ( 1 - 34 T + 4943 T^{2} - 109308 T^{3} + 4943 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( 1 - 6834 T^{2} + 22234799 T^{4} - 48099402332 T^{6} + 22234799 p^{4} T^{8} - 6834 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( ( 1 - 134 T + 11451 T^{2} - 614228 T^{3} + 11451 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 53 | \( ( 1 - 32 T + 2459 T^{2} + 44544 T^{3} + 2459 p^{2} T^{4} - 32 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( ( 1 - 180 T + 20411 T^{2} - 1412584 T^{3} + 20411 p^{2} T^{4} - 180 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 16362 T^{2} + 130073663 T^{4} - 611308223948 T^{6} + 130073663 p^{4} T^{8} - 16362 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( 1 - 12754 T^{2} + 78010639 T^{4} - 366668429212 T^{6} + 78010639 p^{4} T^{8} - 12754 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 21862 T^{2} + 231858863 T^{4} - 1468469748948 T^{6} + 231858863 p^{4} T^{8} - 21862 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( 1 - 27558 T^{2} + 338289263 T^{4} - 2339843433812 T^{6} + 338289263 p^{4} T^{8} - 27558 p^{8} T^{10} + p^{12} T^{12} \) |
| 79 | \( 1 - 11590 T^{2} + 98296655 T^{4} - 773190011028 T^{6} + 98296655 p^{4} T^{8} - 11590 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( 1 - 17170 T^{2} + 191959631 T^{4} - 1435306239516 T^{6} + 191959631 p^{4} T^{8} - 17170 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( ( 1 + 38 T + 9823 T^{2} + 446996 T^{3} + 9823 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( 1 - 48966 T^{2} + 1047505295 T^{4} - 12708004801940 T^{6} + 1047505295 p^{4} T^{8} - 48966 p^{8} T^{10} + p^{12} T^{12} \) |
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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
−4.82155423076572412103718086350, −4.76896571616131945306549689470, −4.75584868265430379396577372697, −4.18569609139991169348310433943, −4.11090720216897815530790650083, −4.06797122011316538140973190243, −3.86621429970129235614906653088, −3.85752617732724473786958937867, −3.60857991710235021639806646737, −3.52055180228065528383175194747, −3.16827804783826439664183654024, −3.04750364491440946851549622484, −2.77434254112836508878712207093, −2.56773687018452736434320750229, −2.53030477948538587832540220296, −2.28898403926434295542501716099, −2.19047629754779823378937246228, −1.70407882511247889519426705425, −1.69218950264442558229230477410, −1.49629490201635295821409442912, −1.30715032667721752009115615736, −0.991624008725585111627323306909, −0.58453115693454411583018332134, −0.56538060070946042513462071153, −0.44837598788045507279297925579,
0.44837598788045507279297925579, 0.56538060070946042513462071153, 0.58453115693454411583018332134, 0.991624008725585111627323306909, 1.30715032667721752009115615736, 1.49629490201635295821409442912, 1.69218950264442558229230477410, 1.70407882511247889519426705425, 2.19047629754779823378937246228, 2.28898403926434295542501716099, 2.53030477948538587832540220296, 2.56773687018452736434320750229, 2.77434254112836508878712207093, 3.04750364491440946851549622484, 3.16827804783826439664183654024, 3.52055180228065528383175194747, 3.60857991710235021639806646737, 3.85752617732724473786958937867, 3.86621429970129235614906653088, 4.06797122011316538140973190243, 4.11090720216897815530790650083, 4.18569609139991169348310433943, 4.75584868265430379396577372697, 4.76896571616131945306549689470, 4.82155423076572412103718086350
Plot not available for L-functions of degree greater than 10.