| L(s) = 1 | − 6·3-s + 2·5-s − 6·7-s + 21·9-s − 3·11-s − 12·15-s + 6·17-s − 3·19-s + 36·21-s − 6·23-s − 8·25-s − 56·27-s + 5·29-s − 4·31-s + 18·33-s − 12·35-s + 37-s + 12·41-s − 6·43-s + 42·45-s − 6·47-s + 21·49-s − 36·51-s + 10·53-s − 6·55-s + 18·57-s − 14·59-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 0.894·5-s − 2.26·7-s + 7·9-s − 0.904·11-s − 3.09·15-s + 1.45·17-s − 0.688·19-s + 7.85·21-s − 1.25·23-s − 8/5·25-s − 10.7·27-s + 0.928·29-s − 0.718·31-s + 3.13·33-s − 2.02·35-s + 0.164·37-s + 1.87·41-s − 0.914·43-s + 6.26·45-s − 0.875·47-s + 3·49-s − 5.04·51-s + 1.37·53-s − 0.809·55-s + 2.38·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013219729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013219729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{6} \) |
| 7 | \( ( 1 + T )^{6} \) |
| 23 | \( ( 1 + T )^{6} \) |
| good | 5 | \( 1 - 2 T + 12 T^{2} - 33 T^{3} + 107 T^{4} - 233 T^{5} + 704 T^{6} - 233 p T^{7} + 107 p^{2} T^{8} - 33 p^{3} T^{9} + 12 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 3 T + 25 T^{2} + 90 T^{3} + 571 T^{4} + 1571 T^{5} + 6806 T^{6} + 1571 p T^{7} + 571 p^{2} T^{8} + 90 p^{3} T^{9} + 25 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 12 T^{2} - 11 T^{3} + 197 T^{4} - 337 T^{5} + 3196 T^{6} - 337 p T^{7} + 197 p^{2} T^{8} - 11 p^{3} T^{9} + 12 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T + 39 T^{2} - 226 T^{3} + 1303 T^{4} - 6056 T^{5} + 24418 T^{6} - 6056 p T^{7} + 1303 p^{2} T^{8} - 226 p^{3} T^{9} + 39 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T + 73 T^{2} + 210 T^{3} + 2859 T^{4} + 6971 T^{5} + 67286 T^{6} + 6971 p T^{7} + 2859 p^{2} T^{8} + 210 p^{3} T^{9} + 73 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 5 T + 99 T^{2} - 544 T^{3} + 5289 T^{4} - 27923 T^{5} + 6318 p T^{6} - 27923 p T^{7} + 5289 p^{2} T^{8} - 544 p^{3} T^{9} + 99 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 4 T + 49 T^{2} + 8 T^{3} + 415 T^{4} - 10892 T^{5} - 20898 T^{6} - 10892 p T^{7} + 415 p^{2} T^{8} + 8 p^{3} T^{9} + 49 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - T + 159 T^{2} - 28 T^{3} + 11545 T^{4} + 2949 T^{5} + 14054 p T^{6} + 2949 p T^{7} + 11545 p^{2} T^{8} - 28 p^{3} T^{9} + 159 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 12 T + 132 T^{2} - 652 T^{3} + 3972 T^{4} - 8476 T^{5} + 87366 T^{6} - 8476 p T^{7} + 3972 p^{2} T^{8} - 652 p^{3} T^{9} + 132 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 6 T + 128 T^{2} + 571 T^{3} + 8211 T^{4} + 26493 T^{5} + 385032 T^{6} + 26493 p T^{7} + 8211 p^{2} T^{8} + 571 p^{3} T^{9} + 128 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 6 T + 209 T^{2} + 800 T^{3} + 18535 T^{4} + 48802 T^{5} + 1031438 T^{6} + 48802 p T^{7} + 18535 p^{2} T^{8} + 800 p^{3} T^{9} + 209 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 10 T + 256 T^{2} - 37 p T^{3} + 27047 T^{4} - 169209 T^{5} + 1725728 T^{6} - 169209 p T^{7} + 27047 p^{2} T^{8} - 37 p^{4} T^{9} + 256 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 14 T + 298 T^{2} + 3351 T^{3} + 37621 T^{4} + 351297 T^{5} + 2779616 T^{6} + 351297 p T^{7} + 37621 p^{2} T^{8} + 3351 p^{3} T^{9} + 298 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 4 T + 272 T^{2} - 953 T^{3} + 35101 T^{4} - 104871 T^{5} + 2702852 T^{6} - 104871 p T^{7} + 35101 p^{2} T^{8} - 953 p^{3} T^{9} + 272 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 7 T + 213 T^{2} - 1641 T^{3} + 28203 T^{4} - 178550 T^{5} + 2328126 T^{6} - 178550 p T^{7} + 28203 p^{2} T^{8} - 1641 p^{3} T^{9} + 213 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 7 T + 255 T^{2} - 1599 T^{3} + 34771 T^{4} - 192504 T^{5} + 3014298 T^{6} - 192504 p T^{7} + 34771 p^{2} T^{8} - 1599 p^{3} T^{9} + 255 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 10 T + 235 T^{2} - 946 T^{3} + 18759 T^{4} + 2748 T^{5} + 1094522 T^{6} + 2748 p T^{7} + 18759 p^{2} T^{8} - 946 p^{3} T^{9} + 235 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 14 T + 437 T^{2} - 4660 T^{3} + 81019 T^{4} - 669726 T^{5} + 8334494 T^{6} - 669726 p T^{7} + 81019 p^{2} T^{8} - 4660 p^{3} T^{9} + 437 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 14 T + 277 T^{2} - 2124 T^{3} + 31819 T^{4} - 202230 T^{5} + 2904830 T^{6} - 202230 p T^{7} + 31819 p^{2} T^{8} - 2124 p^{3} T^{9} + 277 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 25 T + 541 T^{2} - 7201 T^{3} + 90491 T^{4} - 869630 T^{5} + 8970830 T^{6} - 869630 p T^{7} + 90491 p^{2} T^{8} - 7201 p^{3} T^{9} + 541 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 11 T + 355 T^{2} - 1540 T^{3} + 38305 T^{4} + 70111 T^{5} + 2600022 T^{6} + 70111 p T^{7} + 38305 p^{2} T^{8} - 1540 p^{3} T^{9} + 355 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.58448922537423100197836031653, −4.04666514602237599920036993798, −4.01091649848217638750655751420, −3.95962724483342766351098881158, −3.91838058503412446583292277458, −3.89003208791726907561584942359, −3.60888989527324316014901690336, −3.37718664940159622149640016503, −3.12574135764051552719960592077, −3.03033486058376197145601765755, −2.98173396778185309543186358915, −2.75894432622695631451642367556, −2.75838947502701158687289080106, −2.05874050617516653034354634267, −2.02927711936346008204352780018, −1.96004577545036188137300949957, −1.93840447342989340304552813817, −1.92243401513003877793563418370, −1.64019509551910927463640065731, −0.933902714068986037982590646406, −0.932023196189321971144238877995, −0.816634472394409674566550470649, −0.59072176155839708179094970443, −0.45584450918207657264681706070, −0.23686471058782269066763689600,
0.23686471058782269066763689600, 0.45584450918207657264681706070, 0.59072176155839708179094970443, 0.816634472394409674566550470649, 0.932023196189321971144238877995, 0.933902714068986037982590646406, 1.64019509551910927463640065731, 1.92243401513003877793563418370, 1.93840447342989340304552813817, 1.96004577545036188137300949957, 2.02927711936346008204352780018, 2.05874050617516653034354634267, 2.75838947502701158687289080106, 2.75894432622695631451642367556, 2.98173396778185309543186358915, 3.03033486058376197145601765755, 3.12574135764051552719960592077, 3.37718664940159622149640016503, 3.60888989527324316014901690336, 3.89003208791726907561584942359, 3.91838058503412446583292277458, 3.95962724483342766351098881158, 4.01091649848217638750655751420, 4.04666514602237599920036993798, 4.58448922537423100197836031653
Plot not available for L-functions of degree greater than 10.
