| L(s) = 1 | + 6·3-s − 6·5-s + 3·7-s + 21·9-s + 3·11-s + 3·13-s − 36·15-s + 6·17-s − 3·19-s + 18·21-s + 6·23-s + 21·25-s + 56·27-s + 21·29-s − 3·31-s + 18·33-s − 18·35-s + 18·39-s + 9·41-s − 3·43-s − 126·45-s + 21·47-s − 15·49-s + 36·51-s + 15·53-s − 18·55-s − 18·57-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 2.68·5-s + 1.13·7-s + 7·9-s + 0.904·11-s + 0.832·13-s − 9.29·15-s + 1.45·17-s − 0.688·19-s + 3.92·21-s + 1.25·23-s + 21/5·25-s + 10.7·27-s + 3.89·29-s − 0.538·31-s + 3.13·33-s − 3.04·35-s + 2.88·39-s + 1.40·41-s − 0.457·43-s − 18.7·45-s + 3.06·47-s − 2.14·49-s + 5.04·51-s + 2.06·53-s − 2.42·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(119.1449151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(119.1449151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( ( 1 + T )^{6} \) |
| 67 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 3 T + 24 T^{2} - 45 T^{3} + 282 T^{4} - 474 T^{5} + 2458 T^{6} - 474 p T^{7} + 282 p^{2} T^{8} - 45 p^{3} T^{9} + 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 3 T + 30 T^{2} - 75 T^{3} + 348 T^{4} - 876 T^{5} + 3130 T^{6} - 876 p T^{7} + 348 p^{2} T^{8} - 75 p^{3} T^{9} + 30 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 3 T + 3 p T^{2} - 76 T^{3} + 648 T^{4} - 639 T^{5} + 8106 T^{6} - 639 p T^{7} + 648 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{5} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T + 78 T^{2} - 385 T^{3} + 2724 T^{4} - 11127 T^{5} + 57328 T^{6} - 11127 p T^{7} + 2724 p^{2} T^{8} - 385 p^{3} T^{9} + 78 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T + 81 T^{2} + 258 T^{3} + 3060 T^{4} + 9339 T^{5} + 71452 T^{6} + 9339 p T^{7} + 3060 p^{2} T^{8} + 258 p^{3} T^{9} + 81 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 6 T + 78 T^{2} - 436 T^{3} + 3354 T^{4} - 16590 T^{5} + 93454 T^{6} - 16590 p T^{7} + 3354 p^{2} T^{8} - 436 p^{3} T^{9} + 78 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 21 T + 249 T^{2} - 2108 T^{3} + 14766 T^{4} - 90339 T^{5} + 507904 T^{6} - 90339 p T^{7} + 14766 p^{2} T^{8} - 2108 p^{3} T^{9} + 249 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 3 T + 114 T^{2} + 209 T^{3} + 6642 T^{4} + 10098 T^{5} + 255648 T^{6} + 10098 p T^{7} + 6642 p^{2} T^{8} + 209 p^{3} T^{9} + 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 120 T^{2} - 180 T^{3} + 7944 T^{4} - 10224 T^{5} + 370078 T^{6} - 10224 p T^{7} + 7944 p^{2} T^{8} - 180 p^{3} T^{9} + 120 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 9 T + 186 T^{2} - 1421 T^{3} + 16128 T^{4} - 102162 T^{5} + 834046 T^{6} - 102162 p T^{7} + 16128 p^{2} T^{8} - 1421 p^{3} T^{9} + 186 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T + 90 T^{2} + 609 T^{3} + 6102 T^{4} + 34602 T^{5} + 344926 T^{6} + 34602 p T^{7} + 6102 p^{2} T^{8} + 609 p^{3} T^{9} + 90 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 21 T + 303 T^{2} - 3044 T^{3} + 27474 T^{4} - 213243 T^{5} + 1570804 T^{6} - 213243 p T^{7} + 27474 p^{2} T^{8} - 3044 p^{3} T^{9} + 303 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 15 T + 147 T^{2} - 1020 T^{3} + 8466 T^{4} - 1461 p T^{5} + 656404 T^{6} - 1461 p^{2} T^{7} + 8466 p^{2} T^{8} - 1020 p^{3} T^{9} + 147 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 15 T + 345 T^{2} - 3492 T^{3} + 46986 T^{4} - 360735 T^{5} + 3571414 T^{6} - 360735 p T^{7} + 46986 p^{2} T^{8} - 3492 p^{3} T^{9} + 345 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 15 T + 333 T^{2} - 3566 T^{3} + 46368 T^{4} - 386895 T^{5} + 3658196 T^{6} - 386895 p T^{7} + 46368 p^{2} T^{8} - 3566 p^{3} T^{9} + 333 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 300 T^{2} - 3057 T^{3} + 34512 T^{4} - 321813 T^{5} + 3065020 T^{6} - 321813 p T^{7} + 34512 p^{2} T^{8} - 3057 p^{3} T^{9} + 300 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 6 T + 228 T^{2} + 1709 T^{3} + 28116 T^{4} + 226017 T^{5} + 2344662 T^{6} + 226017 p T^{7} + 28116 p^{2} T^{8} + 1709 p^{3} T^{9} + 228 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 9 T + 84 T^{2} - 1001 T^{3} + 2028 T^{4} + 27336 T^{5} + 181484 T^{6} + 27336 p T^{7} + 2028 p^{2} T^{8} - 1001 p^{3} T^{9} + 84 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 24 T + 708 T^{2} - 10769 T^{3} + 174270 T^{4} - 1854945 T^{5} + 20317210 T^{6} - 1854945 p T^{7} + 174270 p^{2} T^{8} - 10769 p^{3} T^{9} + 708 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 24 T + 276 T^{2} - 1213 T^{3} - 5016 T^{4} + 173625 T^{5} - 1927802 T^{6} + 173625 p T^{7} - 5016 p^{2} T^{8} - 1213 p^{3} T^{9} + 276 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 9 T + 453 T^{2} - 2388 T^{3} + 81744 T^{4} - 252837 T^{5} + 9145678 T^{6} - 252837 p T^{7} + 81744 p^{2} T^{8} - 2388 p^{3} T^{9} + 453 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.36434429129745148811402854217, −4.01889257690425157284507114580, −3.87837823957319339731846063190, −3.82187588336688926904958804231, −3.76402891205986789806539529261, −3.75256100135535033529430532951, −3.53642447442153771106529561847, −3.20982438232643432281092794971, −3.13370909646083734329401925162, −3.06615577347969661756512045203, −3.05910331330439458177127636169, −2.81470225961021442085179187972, −2.80678235765138250340047424028, −2.26389373410148696314742427204, −2.20630203692162372933231207013, −2.10223663885880934064809269300, −2.06306967510104273748995188417, −1.86769401053889291804268499106, −1.76687251632634790541224133387, −1.09834383463526026182332243961, −1.04609504354782529066400957515, −0.906174268822109083688411882235, −0.799204878053610257137096012836, −0.78194812122568250499648278787, −0.60424608790646362532744039729,
0.60424608790646362532744039729, 0.78194812122568250499648278787, 0.799204878053610257137096012836, 0.906174268822109083688411882235, 1.04609504354782529066400957515, 1.09834383463526026182332243961, 1.76687251632634790541224133387, 1.86769401053889291804268499106, 2.06306967510104273748995188417, 2.10223663885880934064809269300, 2.20630203692162372933231207013, 2.26389373410148696314742427204, 2.80678235765138250340047424028, 2.81470225961021442085179187972, 3.05910331330439458177127636169, 3.06615577347969661756512045203, 3.13370909646083734329401925162, 3.20982438232643432281092794971, 3.53642447442153771106529561847, 3.75256100135535033529430532951, 3.76402891205986789806539529261, 3.82187588336688926904958804231, 3.87837823957319339731846063190, 4.01889257690425157284507114580, 4.36434429129745148811402854217
Plot not available for L-functions of degree greater than 10.
