| L(s) = 1 | − 3·2-s − 9·3-s + 5·4-s + 15·5-s + 27·6-s − 6·8-s + 27·9-s − 45·10-s + 9·11-s − 45·12-s + 11·13-s − 135·15-s + 7·16-s − 81·18-s + 38·19-s + 75·20-s − 27·22-s − 15·23-s + 54·24-s + 62·25-s − 33·26-s + 54·27-s − 51·29-s + 405·30-s + 46·31-s + 33·32-s − 81·33-s + ⋯ |
| L(s) = 1 | − 3/2·2-s − 3·3-s + 5/4·4-s + 3·5-s + 9/2·6-s − 3/4·8-s + 3·9-s − 9/2·10-s + 9/11·11-s − 3.75·12-s + 0.846·13-s − 9·15-s + 7/16·16-s − 9/2·18-s + 2·19-s + 15/4·20-s − 1.22·22-s − 0.652·23-s + 9/4·24-s + 2.47·25-s − 1.26·26-s + 2·27-s − 1.75·29-s + 27/2·30-s + 1.48·31-s + 1.03·32-s − 2.45·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.035946652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035946652\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{2} T^{2} + 3 T^{3} - 45 T^{5} - 169 T^{6} - 45 p^{2} T^{7} + 3 p^{6} T^{9} + p^{10} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \) |
| 5 | \( 1 - 3 p T + 163 T^{2} - 264 p T^{3} + 9249 T^{4} - 55833 T^{5} + 298502 T^{6} - 55833 p^{2} T^{7} + 9249 p^{4} T^{8} - 264 p^{7} T^{9} + 163 p^{8} T^{10} - 3 p^{11} T^{11} + p^{12} T^{12} \) |
| 11 | \( 1 - 9 T + 175 T^{2} - 1332 T^{3} + 3213 T^{4} + 170397 T^{5} - 1265338 T^{6} + 170397 p^{2} T^{7} + 3213 p^{4} T^{8} - 1332 p^{6} T^{9} + 175 p^{8} T^{10} - 9 p^{10} T^{11} + p^{12} T^{12} \) |
| 13 | \( 1 - 11 T - 25 p T^{2} + 1116 T^{3} + 90421 T^{4} + 1151 T^{5} - 18352946 T^{6} + 1151 p^{2} T^{7} + 90421 p^{4} T^{8} + 1116 p^{6} T^{9} - 25 p^{9} T^{10} - 11 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( 1 - 141 T^{2} + 3174 T^{4} + 4877719 T^{6} + 3174 p^{4} T^{8} - 141 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 - p T + 1022 T^{2} - 13255 T^{3} + 1022 p^{2} T^{4} - p^{5} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 + 15 T + 1503 T^{2} + 21420 T^{3} + 1360017 T^{4} + 764259 p T^{5} + 1586350 p^{2} T^{6} + 764259 p^{3} T^{7} + 1360017 p^{4} T^{8} + 21420 p^{6} T^{9} + 1503 p^{8} T^{10} + 15 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( 1 + 51 T + 3667 T^{2} + 142800 T^{3} + 6968025 T^{4} + 208158669 T^{5} + 7326647606 T^{6} + 208158669 p^{2} T^{7} + 6968025 p^{4} T^{8} + 142800 p^{6} T^{9} + 3667 p^{8} T^{10} + 51 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 46 T - 443 T^{2} + 49846 T^{3} + 235402 T^{4} - 30628678 T^{5} + 546835085 T^{6} - 30628678 p^{2} T^{7} + 235402 p^{4} T^{8} + 49846 p^{6} T^{9} - 443 p^{8} T^{10} - 46 p^{10} T^{11} + p^{12} T^{12} \) |
| 37 | \( ( 1 + 7 T + 1926 T^{2} - 4585 T^{3} + 1926 p^{2} T^{4} + 7 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 + 27 T + 2791 T^{2} + 68796 T^{3} + 2445501 T^{4} - 9372327 T^{5} + 956644694 T^{6} - 9372327 p^{2} T^{7} + 2445501 p^{4} T^{8} + 68796 p^{6} T^{9} + 2791 p^{8} T^{10} + 27 p^{10} T^{11} + p^{12} T^{12} \) |
| 43 | \( 1 + 99 T + 2211 T^{2} + 60788 T^{3} + 9388401 T^{4} + 180657609 T^{5} - 8142632682 T^{6} + 180657609 p^{2} T^{7} + 9388401 p^{4} T^{8} + 60788 p^{6} T^{9} + 2211 p^{8} T^{10} + 99 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 - 156 T + 13731 T^{2} - 876564 T^{3} + 39589302 T^{4} - 1446227628 T^{5} + 60047988631 T^{6} - 1446227628 p^{2} T^{7} + 39589302 p^{4} T^{8} - 876564 p^{6} T^{9} + 13731 p^{8} T^{10} - 156 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 - 10581 T^{2} + 59433918 T^{4} - 206108979833 T^{6} + 59433918 p^{4} T^{8} - 10581 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 + 144 T + 18363 T^{2} + 1648944 T^{3} + 137014998 T^{4} + 9169460352 T^{5} + 591153999199 T^{6} + 9169460352 p^{2} T^{7} + 137014998 p^{4} T^{8} + 1648944 p^{6} T^{9} + 18363 p^{8} T^{10} + 144 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 + 22 T - 5683 T^{2} + 129618 T^{3} + 15017866 T^{4} - 686403730 T^{5} - 45168128507 T^{6} - 686403730 p^{2} T^{7} + 15017866 p^{4} T^{8} + 129618 p^{6} T^{9} - 5683 p^{8} T^{10} + 22 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 - 98 T - 2707 T^{2} + 132426 T^{3} + 30690346 T^{4} + 615326054 T^{5} - 244804698011 T^{6} + 615326054 p^{2} T^{7} + 30690346 p^{4} T^{8} + 132426 p^{6} T^{9} - 2707 p^{8} T^{10} - 98 p^{10} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 - 24678 T^{2} + 270556335 T^{4} - 1729620874964 T^{6} + 270556335 p^{4} T^{8} - 24678 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 101 T + 8022 T^{2} - 267985 T^{3} + 8022 p^{2} T^{4} - 101 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 + 90 T - 12891 T^{2} - 390994 T^{3} + 207008970 T^{4} + 3856308882 T^{5} - 1279433384115 T^{6} + 3856308882 p^{2} T^{7} + 207008970 p^{4} T^{8} - 390994 p^{6} T^{9} - 12891 p^{8} T^{10} + 90 p^{10} T^{11} + p^{12} T^{12} \) |
| 83 | \( 1 - 99 T + 14007 T^{2} - 1063260 T^{3} + 63095685 T^{4} - 2375688321 T^{5} + 130315729030 T^{6} - 2375688321 p^{2} T^{7} + 63095685 p^{4} T^{8} - 1063260 p^{6} T^{9} + 14007 p^{8} T^{10} - 99 p^{10} T^{11} + p^{12} T^{12} \) |
| 89 | \( 1 - 34229 T^{2} + 565360374 T^{4} - 5639336113825 T^{6} + 565360374 p^{4} T^{8} - 34229 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( 1 + 161 T - 5957 T^{2} - 542804 T^{3} + 259416241 T^{4} + 8128908515 T^{5} - 2138448122938 T^{6} + 8128908515 p^{2} T^{7} + 259416241 p^{4} T^{8} - 542804 p^{6} T^{9} - 5957 p^{8} T^{10} + 161 p^{10} T^{11} + p^{12} T^{12} \) |
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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
−5.95992518681798198724384157869, −5.54901978986687286059240493040, −5.43171127279765687093041428138, −5.28697949047018819268136406794, −5.22104197273321213402412886637, −5.21696761729214824449407182684, −5.12801528269360011010212824589, −4.57454968286698196842113730955, −4.45079325191514820960238407551, −4.08564737868857959411474558963, −3.96558199055940191286031495939, −3.88512204868138073254390314535, −3.19771014943896372584735645390, −3.16841693616635937435350705890, −3.11351820426168298940621642882, −2.77131965380038522849741538471, −2.25257712748826754900117061074, −2.25048776902452066979374209907, −1.91804824058168306774095539292, −1.73639705692171209722472753583, −1.27068903934422587136134047257, −1.25519311771019743778782613126, −0.857452274116869655547352218094, −0.60295834855420062487188015015, −0.28951308435949802659408037700,
0.28951308435949802659408037700, 0.60295834855420062487188015015, 0.857452274116869655547352218094, 1.25519311771019743778782613126, 1.27068903934422587136134047257, 1.73639705692171209722472753583, 1.91804824058168306774095539292, 2.25048776902452066979374209907, 2.25257712748826754900117061074, 2.77131965380038522849741538471, 3.11351820426168298940621642882, 3.16841693616635937435350705890, 3.19771014943896372584735645390, 3.88512204868138073254390314535, 3.96558199055940191286031495939, 4.08564737868857959411474558963, 4.45079325191514820960238407551, 4.57454968286698196842113730955, 5.12801528269360011010212824589, 5.21696761729214824449407182684, 5.22104197273321213402412886637, 5.28697949047018819268136406794, 5.43171127279765687093041428138, 5.54901978986687286059240493040, 5.95992518681798198724384157869
Plot not available for L-functions of degree greater than 10.
