| L(s) = 1 | − 4·5-s + 8·9-s + 4·13-s + 8·25-s − 8·29-s − 52·37-s + 36·41-s − 32·45-s + 8·53-s + 56·61-s − 16·65-s + 4·73-s + 16·81-s + 20·89-s − 4·97-s − 36·109-s − 56·113-s + 32·117-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 8/3·9-s + 1.10·13-s + 8/5·25-s − 1.48·29-s − 8.54·37-s + 5.62·41-s − 4.77·45-s + 1.09·53-s + 7.17·61-s − 1.98·65-s + 0.468·73-s + 16/9·81-s + 2.11·89-s − 0.406·97-s − 3.44·109-s − 5.26·113-s + 2.95·117-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8248816844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8248816844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - 2 T + 21 T^{2} - 64 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 16 T^{4} - 22 p T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 + 2 T + 2 T^{2} - 14 T^{3} - 24 T^{4} + 46 T^{5} + 238 T^{6} + 46 p T^{7} - 24 p^{2} T^{8} - 14 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 - 260 T^{4} + 29672 T^{8} - 1894058 T^{12} + 29672 p^{4} T^{16} - 260 p^{8} T^{20} + p^{12} T^{24} \) |
| 11 | \( ( 1 + 14 T^{4} + p^{4} T^{8} )^{3} \) |
| 17 | \( ( 1 - 4 p T^{2} + 2344 T^{4} - 49642 T^{6} + 2344 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( 1 + 466 T^{4} - 96529 T^{8} - 114854372 T^{12} - 96529 p^{4} T^{16} + 466 p^{8} T^{20} + p^{12} T^{24} \) |
| 23 | \( ( 1 + 6 T^{2} + 1011 T^{4} + 2284 T^{6} + 1011 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 2 T + 33 T^{2} + 224 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 31 | \( 1 + 1282 T^{4} + 1391999 T^{8} + 999051772 T^{12} + 1391999 p^{4} T^{16} + 1282 p^{8} T^{20} + p^{12} T^{24} \) |
| 37 | \( ( 1 + 26 T + 338 T^{2} + 3386 T^{3} + 29960 T^{4} + 224630 T^{5} + 1446398 T^{6} + 224630 p T^{7} + 29960 p^{2} T^{8} + 3386 p^{3} T^{9} + 338 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 18 T + 162 T^{2} - 1026 T^{3} + 7155 T^{4} - 59652 T^{5} + 440964 T^{6} - 59652 p T^{7} + 7155 p^{2} T^{8} - 1026 p^{3} T^{9} + 162 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 208 T^{2} + 19544 T^{4} + 1070182 T^{6} + 19544 p^{2} T^{8} + 208 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 9156 T^{4} + 40728552 T^{8} - 111343409098 T^{12} + 40728552 p^{4} T^{16} - 9156 p^{8} T^{20} + p^{12} T^{24} \) |
| 53 | \( ( 1 - 2 T + 141 T^{2} - 224 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( 1 - 7182 T^{4} + 48374415 T^{8} - 178939735972 T^{12} + 48374415 p^{4} T^{16} - 7182 p^{8} T^{20} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 14 T + 171 T^{2} - 1252 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( 1 + 21802 T^{4} + 211748159 T^{8} + 1203289195852 T^{12} + 211748159 p^{4} T^{16} + 21802 p^{8} T^{20} + p^{12} T^{24} \) |
| 71 | \( 1 - 12900 T^{4} + 109295880 T^{8} - 644604200746 T^{12} + 109295880 p^{4} T^{16} - 12900 p^{8} T^{20} + p^{12} T^{24} \) |
| 73 | \( ( 1 - 2 T + 2 T^{2} - 50 T^{3} + 7667 T^{4} - 22628 T^{5} + 31172 T^{6} - 22628 p T^{7} + 7667 p^{2} T^{8} - 50 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 226 T^{2} + 26243 T^{4} - 2222788 T^{6} + 26243 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 1234 T^{4} + p^{4} T^{8} )^{3} \) |
| 89 | \( ( 1 - 10 T + 50 T^{2} + 22 T^{3} + 915 T^{4} - 96836 T^{5} + 922852 T^{6} - 96836 p T^{7} + 915 p^{2} T^{8} + 22 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 2 T + 2 T^{2} + 242 T^{3} + 14927 T^{4} - 1108 T^{5} - 2788 T^{6} - 1108 p T^{7} + 14927 p^{2} T^{8} + 242 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.44858875990143128798001267960, −3.29548371269369752672934939184, −3.12098094102295477695830215636, −2.97878687238501600706556643299, −2.93041082546688393326943785102, −2.92025404208160605370723220033, −2.74811279175868818154952577882, −2.45716836259919176275741423861, −2.41718701299141496211148787069, −2.36224996354585817423770874495, −2.24001719197669288145761368295, −2.15867678611163508539841537784, −2.01882470471095514328301073342, −1.88355664826588330981076232246, −1.81795611325898841663051474311, −1.61763159098879467368958840290, −1.57809467879946459902918216368, −1.40645843797963306286480954605, −1.23812615827451829324608107219, −1.02368060066087857420636623495, −0.947085123047646141583614823579, −0.942466508687522768420281370314, −0.62515096674347538960783363056, −0.39859565019557671456132282268, −0.086203401709749216790304934028,
0.086203401709749216790304934028, 0.39859565019557671456132282268, 0.62515096674347538960783363056, 0.942466508687522768420281370314, 0.947085123047646141583614823579, 1.02368060066087857420636623495, 1.23812615827451829324608107219, 1.40645843797963306286480954605, 1.57809467879946459902918216368, 1.61763159098879467368958840290, 1.81795611325898841663051474311, 1.88355664826588330981076232246, 2.01882470471095514328301073342, 2.15867678611163508539841537784, 2.24001719197669288145761368295, 2.36224996354585817423770874495, 2.41718701299141496211148787069, 2.45716836259919176275741423861, 2.74811279175868818154952577882, 2.92025404208160605370723220033, 2.93041082546688393326943785102, 2.97878687238501600706556643299, 3.12098094102295477695830215636, 3.29548371269369752672934939184, 3.44858875990143128798001267960
Plot not available for L-functions of degree greater than 10.
