| L(s) = 1 | − 5·4-s − 5·9-s + 2·11-s + 15·16-s + 12·29-s − 30·31-s + 25·36-s − 12·41-s − 10·44-s − 20·49-s − 20·59-s − 2·61-s − 40·64-s + 2·71-s − 24·79-s + 10·81-s − 36·89-s − 10·99-s + 14·101-s + 50·109-s − 60·116-s − 55·121-s + 150·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 5/2·4-s − 5/3·9-s + 0.603·11-s + 15/4·16-s + 2.22·29-s − 5.38·31-s + 25/6·36-s − 1.87·41-s − 1.50·44-s − 2.85·49-s − 2.60·59-s − 0.256·61-s − 5·64-s + 0.237·71-s − 2.70·79-s + 10/9·81-s − 3.81·89-s − 1.00·99-s + 1.39·101-s + 4.78·109-s − 5.57·116-s − 5·121-s + 13.4·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 5 T^{2} + 5 p T^{4} + 15 T^{6} + 5 p^{3} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 + 5 T^{2} + 5 p T^{4} + 29 T^{6} + 5 p^{3} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 20 T^{2} + 22 p T^{4} + 873 T^{6} + 22 p^{3} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 29 T^{2} - 19 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 47 T^{2} + 1162 T^{4} + 18711 T^{6} + 1162 p^{2} T^{8} + 47 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 67 T^{2} + 2153 T^{4} + 44221 T^{6} + 2153 p^{2} T^{8} + 67 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 126 T^{2} + 6838 T^{4} + 205573 T^{6} + 6838 p^{2} T^{8} + 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + 60 T^{2} - 321 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 15 T + 163 T^{2} + 1027 T^{3} + 163 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 124 T^{2} + 6918 T^{4} + 272997 T^{6} + 6918 p^{2} T^{8} + 124 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 2 p T^{2} + 489 T^{3} + 2 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 131 T^{2} + 8407 T^{4} + 397137 T^{6} + 8407 p^{2} T^{8} + 131 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 68 T^{2} + 6310 T^{4} + 282273 T^{6} + 6310 p^{2} T^{8} + 68 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 219 T^{2} + 23449 T^{4} + 1541557 T^{6} + 23449 p^{2} T^{8} + 219 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 10 T + 122 T^{2} + 889 T^{3} + 122 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + T + 142 T^{2} + 9 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 326 T^{2} + 48823 T^{4} + 4202580 T^{6} + 48823 p^{2} T^{8} + 326 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - T + 89 T^{2} - 619 T^{3} + 89 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 202 T^{2} + 24354 T^{4} + 2081577 T^{6} + 24354 p^{2} T^{8} + 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 205 T^{2} + 1448 T^{3} + 205 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 39 T^{2} + 7249 T^{4} + 843277 T^{6} + 7249 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 18 T + 295 T^{2} + 3132 T^{3} + 295 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 53 T^{2} + 10075 T^{4} + 1543773 T^{6} + 10075 p^{2} T^{8} + 53 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.42178988759466667236073036611, −4.15368379720016700921155003243, −4.00868864084842035494349953023, −3.88041056455957853822311531056, −3.70333175257905226007038587726, −3.69913781035572297535308704698, −3.59933211270721734681109248698, −3.33972956835850875894087021456, −3.23197037729089382127218849069, −3.17078232741772619209734014900, −3.15679337781192121708670486926, −3.13539898880615646591987204513, −2.61799672239171347448356856167, −2.60014833612717651113545749361, −2.50080170859200979975754577195, −2.28837768492497464192199623668, −2.18248730467497490524401181508, −1.92031339426973599159852261100, −1.62099962742978483621219732820, −1.61868294201950829488712502428, −1.52060297327769119375047872592, −1.24945860624362460930496647862, −1.12179895785983688750583963102, −0.985692701091211987085546687514, −0.917805387583713856093890839137, 0, 0, 0, 0, 0, 0,
0.917805387583713856093890839137, 0.985692701091211987085546687514, 1.12179895785983688750583963102, 1.24945860624362460930496647862, 1.52060297327769119375047872592, 1.61868294201950829488712502428, 1.62099962742978483621219732820, 1.92031339426973599159852261100, 2.18248730467497490524401181508, 2.28837768492497464192199623668, 2.50080170859200979975754577195, 2.60014833612717651113545749361, 2.61799672239171347448356856167, 3.13539898880615646591987204513, 3.15679337781192121708670486926, 3.17078232741772619209734014900, 3.23197037729089382127218849069, 3.33972956835850875894087021456, 3.59933211270721734681109248698, 3.69913781035572297535308704698, 3.70333175257905226007038587726, 3.88041056455957853822311531056, 4.00868864084842035494349953023, 4.15368379720016700921155003243, 4.42178988759466667236073036611
Plot not available for L-functions of degree greater than 10.
