| L(s) = 1 | − 3·3-s + 8-s + 3·9-s + 12·11-s + 6·13-s + 6·17-s + 6·19-s + 6·23-s − 3·24-s + 27-s − 12·29-s + 12·31-s − 36·33-s + 24·37-s − 18·39-s + 27·41-s − 30·43-s + 24·47-s + 9·49-s − 18·51-s + 12·53-s − 18·57-s + 3·59-s − 12·61-s + 27·67-s − 18·69-s + 24·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.353·8-s + 9-s + 3.61·11-s + 1.66·13-s + 1.45·17-s + 1.37·19-s + 1.25·23-s − 0.612·24-s + 0.192·27-s − 2.22·29-s + 2.15·31-s − 6.26·33-s + 3.94·37-s − 2.88·39-s + 4.21·41-s − 4.57·43-s + 3.50·47-s + 9/7·49-s − 2.52·51-s + 1.64·53-s − 2.38·57-s + 0.390·59-s − 1.53·61-s + 3.29·67-s − 2.16·69-s + 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.675331006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.675331006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 + p T + 2 p T^{2} + 8 T^{3} + 7 p T^{4} + 17 p T^{5} + 109 T^{6} + 17 p^{2} T^{7} + 7 p^{3} T^{8} + 8 p^{3} T^{9} + 2 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 9 T^{2} + 16 T^{3} + 18 T^{4} - 72 T^{5} + 183 T^{6} - 72 p T^{7} + 18 p^{2} T^{8} + 16 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 12 T + 6 p T^{2} - 306 T^{3} + 1446 T^{4} - 5430 T^{5} + 17539 T^{6} - 5430 p T^{7} + 1446 p^{2} T^{8} - 306 p^{3} T^{9} + 6 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 6 T + 48 T^{2} - 230 T^{3} + 1332 T^{4} - 4950 T^{5} + 21111 T^{6} - 4950 p T^{7} + 1332 p^{2} T^{8} - 230 p^{3} T^{9} + 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T + 36 T^{2} - 162 T^{3} + 1062 T^{4} - 4542 T^{5} + 20647 T^{6} - 4542 p T^{7} + 1062 p^{2} T^{8} - 162 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 6 T - 36 T^{3} - 252 T^{4} + 60 p T^{5} + 12421 T^{6} + 60 p^{2} T^{7} - 252 p^{2} T^{8} - 36 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 12 T + 144 T^{2} + 1062 T^{3} + 8568 T^{4} + 49404 T^{5} + 307135 T^{6} + 49404 p T^{7} + 8568 p^{2} T^{8} + 1062 p^{3} T^{9} + 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 12 T + 39 T^{2} - 76 T^{3} + 666 T^{4} + 4428 T^{5} - 70113 T^{6} + 4428 p T^{7} + 666 p^{2} T^{8} - 76 p^{3} T^{9} + 39 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 4 T + p T^{2} )^{6} \) |
| 41 | \( 1 - 27 T + 306 T^{2} - 1692 T^{3} - 171 T^{4} + 93519 T^{5} - 880271 T^{6} + 93519 p T^{7} - 171 p^{2} T^{8} - 1692 p^{3} T^{9} + 306 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 30 T + 390 T^{2} + 2524 T^{3} + 864 T^{4} - 147636 T^{5} - 1460739 T^{6} - 147636 p T^{7} + 864 p^{2} T^{8} + 2524 p^{3} T^{9} + 390 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 24 T + 396 T^{2} - 4860 T^{3} + 50220 T^{4} - 430782 T^{5} + 3197917 T^{6} - 430782 p T^{7} + 50220 p^{2} T^{8} - 4860 p^{3} T^{9} + 396 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 12 T + 72 T^{2} - 558 T^{3} + 2484 T^{4} + 4542 T^{5} - 110141 T^{6} + 4542 p T^{7} + 2484 p^{2} T^{8} - 558 p^{3} T^{9} + 72 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 3 T - 18 T^{2} - 792 T^{3} + 387 T^{4} - 579 T^{5} + 581203 T^{6} - 579 p T^{7} + 387 p^{2} T^{8} - 792 p^{3} T^{9} - 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 12 T + 96 T^{2} + 1322 T^{3} + 9108 T^{4} + 67266 T^{5} + 766539 T^{6} + 67266 p T^{7} + 9108 p^{2} T^{8} + 1322 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 27 T + 450 T^{2} - 5222 T^{3} + 46485 T^{4} - 360603 T^{5} + 2744985 T^{6} - 360603 p T^{7} + 46485 p^{2} T^{8} - 5222 p^{3} T^{9} + 450 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 24 T + 396 T^{2} - 5148 T^{3} + 61452 T^{4} - 608478 T^{5} + 5464981 T^{6} - 608478 p T^{7} + 61452 p^{2} T^{8} - 5148 p^{3} T^{9} + 396 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 12 T + 39 T^{2} - 437 T^{3} - 3591 T^{4} + 24273 T^{5} + 305502 T^{6} + 24273 p T^{7} - 3591 p^{2} T^{8} - 437 p^{3} T^{9} + 39 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 18 T + 144 T^{2} - 898 T^{3} - 2412 T^{4} + 1854 p T^{5} - 1692195 T^{6} + 1854 p^{2} T^{7} - 2412 p^{2} T^{8} - 898 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 6 T - 6 T^{2} - 1422 T^{3} - 6744 T^{4} + 34656 T^{5} + 1449727 T^{6} + 34656 p T^{7} - 6744 p^{2} T^{8} - 1422 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 48 T + 936 T^{2} - 8694 T^{3} + 14904 T^{4} + 643704 T^{5} - 9263141 T^{6} + 643704 p T^{7} + 14904 p^{2} T^{8} - 8694 p^{3} T^{9} + 936 p^{4} T^{10} - 48 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 27 T + 324 T^{2} + 1492 T^{3} - 6993 T^{4} - 245403 T^{5} - 2965839 T^{6} - 245403 p T^{7} - 6993 p^{2} T^{8} + 1492 p^{3} T^{9} + 324 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.34471413597201985331606563130, −5.23453606704898976010005785868, −5.07475371948277599806634076077, −5.02295879841093897242216757208, −4.52338261422179946755648728366, −4.47909673155321108762776628733, −4.39852369050662808573588602638, −4.10959021782764418868081812715, −3.84696970358394496291793081789, −3.83886718575228161587351234862, −3.77767195414585000705498801551, −3.56217065169551406960936215847, −3.56045231951559426082885069729, −3.12669326885103388004781938218, −2.80765206523762642311214092362, −2.60843368248389460779388954392, −2.50193423630867682253001166125, −2.18832666744612306286910023479, −2.15575826254469816567246124244, −1.32126804562551901740252325131, −1.27700485415371258243589419934, −1.17195710930025252630496032116, −0.995160498437921818062091050918, −0.977262816110302438878705934557, −0.57320101114721975073790638260,
0.57320101114721975073790638260, 0.977262816110302438878705934557, 0.995160498437921818062091050918, 1.17195710930025252630496032116, 1.27700485415371258243589419934, 1.32126804562551901740252325131, 2.15575826254469816567246124244, 2.18832666744612306286910023479, 2.50193423630867682253001166125, 2.60843368248389460779388954392, 2.80765206523762642311214092362, 3.12669326885103388004781938218, 3.56045231951559426082885069729, 3.56217065169551406960936215847, 3.77767195414585000705498801551, 3.83886718575228161587351234862, 3.84696970358394496291793081789, 4.10959021782764418868081812715, 4.39852369050662808573588602638, 4.47909673155321108762776628733, 4.52338261422179946755648728366, 5.02295879841093897242216757208, 5.07475371948277599806634076077, 5.23453606704898976010005785868, 5.34471413597201985331606563130
Plot not available for L-functions of degree greater than 10.
