| L(s) = 1 | − 2·5-s + 18·13-s + 2·17-s − 17·19-s + 7·25-s + 12·29-s − 4·31-s − 3·41-s − 18·43-s − 18·47-s + 22·49-s + 36·53-s + 27·59-s − 10·61-s − 36·65-s + 11·67-s + 18·71-s − 5·73-s + 16·79-s − 4·85-s + 24·89-s + 34·95-s + 21·97-s + 10·101-s + 4·103-s + 12·109-s + 37·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 4.99·13-s + 0.485·17-s − 3.90·19-s + 7/5·25-s + 2.22·29-s − 0.718·31-s − 0.468·41-s − 2.74·43-s − 2.62·47-s + 22/7·49-s + 4.94·53-s + 3.51·59-s − 1.28·61-s − 4.46·65-s + 1.34·67-s + 2.13·71-s − 0.585·73-s + 1.80·79-s − 0.433·85-s + 2.54·89-s + 3.48·95-s + 2.13·97-s + 0.995·101-s + 0.394·103-s + 1.14·109-s + 3.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.36044671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.36044671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 17 T + 144 T^{2} + 775 T^{3} + 144 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 + 2 T - 3 T^{2} - 14 T^{3} - 14 T^{4} + 8 T^{5} + 101 T^{6} + 8 p T^{7} - 14 p^{2} T^{8} - 14 p^{3} T^{9} - 3 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 22 T^{2} + 275 T^{4} - 2272 T^{6} + 275 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 37 T^{2} + 774 T^{4} - 10229 T^{6} + 774 p^{2} T^{8} - 37 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 - 2 T - 3 T^{2} + 146 T^{3} - 230 T^{4} - 662 T^{5} + 14237 T^{6} - 662 p T^{7} - 230 p^{2} T^{8} + 146 p^{3} T^{9} - 3 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 59 T^{2} + 2124 T^{4} + 1062 T^{5} + 55837 T^{6} + 1062 p T^{7} + 2124 p^{2} T^{8} + 59 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 12 T + 123 T^{2} - 900 T^{3} + 5874 T^{4} - 35358 T^{5} + 196391 T^{6} - 35358 p T^{7} + 5874 p^{2} T^{8} - 900 p^{3} T^{9} + 123 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 2 T + 39 T^{2} + 190 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 150 T^{2} + 10743 T^{4} - 482888 T^{6} + 10743 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 + 3 T + 27 T^{2} + 72 T^{3} - 567 T^{4} - 8859 T^{5} - 110842 T^{6} - 8859 p T^{7} - 567 p^{2} T^{8} + 72 p^{3} T^{9} + 27 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 47 | \( 1 + 18 T + 275 T^{2} + 3006 T^{3} + 30156 T^{4} + 241800 T^{5} + 1812097 T^{6} + 241800 p T^{7} + 30156 p^{2} T^{8} + 3006 p^{3} T^{9} + 275 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 27 T + 339 T^{2} - 3132 T^{3} + 29217 T^{4} - 294057 T^{5} + 2581810 T^{6} - 294057 p T^{7} + 29217 p^{2} T^{8} - 3132 p^{3} T^{9} + 339 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 10 T - 107 T^{2} - 382 T^{3} + 17962 T^{4} + 38452 T^{5} - 1028851 T^{6} + 38452 p T^{7} + 17962 p^{2} T^{8} - 382 p^{3} T^{9} - 107 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 11 T - 41 T^{2} + 176 T^{3} + 4621 T^{4} + 30811 T^{5} - 741550 T^{6} + 30811 p T^{7} + 4621 p^{2} T^{8} + 176 p^{3} T^{9} - 41 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 73 | \( 1 + 5 T - 169 T^{2} - 456 T^{3} + 19909 T^{4} + 23347 T^{5} - 1594250 T^{6} + 23347 p T^{7} + 19909 p^{2} T^{8} - 456 p^{3} T^{9} - 169 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 16 T - 29 T^{2} + 400 T^{3} + 19030 T^{4} - 86176 T^{5} - 877993 T^{6} - 86176 p T^{7} + 19030 p^{2} T^{8} + 400 p^{3} T^{9} - 29 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 385 T^{2} + 68850 T^{4} - 7268141 T^{6} + 68850 p^{2} T^{8} - 385 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 24 T + 387 T^{2} - 4680 T^{3} + 50526 T^{4} - 537636 T^{5} + 5096063 T^{6} - 537636 p T^{7} + 50526 p^{2} T^{8} - 4680 p^{3} T^{9} + 387 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 21 T + 375 T^{2} - 4788 T^{3} + 53661 T^{4} - 577383 T^{5} + 5685046 T^{6} - 577383 p T^{7} + 53661 p^{2} T^{8} - 4788 p^{3} T^{9} + 375 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.72636484814483693695622574735, −4.15643175774707877779518431188, −4.13591851649453379704390425620, −4.13335328543023034725243263650, −3.99199358104512779775218746484, −3.77865172058802319366260122464, −3.64640826041485192048469329655, −3.54912240167799957488974557326, −3.48164638704211024802711859301, −3.43593923365961204052229150795, −3.23128852987003543292519806499, −2.86106982173926784970592032526, −2.82366854276067959334999085121, −2.34543124164208192942882904329, −2.32548911636716198604692855038, −2.25282757288941192961680576392, −2.07031885534292481799669935891, −1.80637773621327526893608400513, −1.70141484667203465306243271462, −1.42765051488170276770502346583, −1.03737881575641143679977197362, −0.974512123645667544407378830245, −0.67627661382497510517764685539, −0.59840224288344752439814348413, −0.51033126592858719357854240819,
0.51033126592858719357854240819, 0.59840224288344752439814348413, 0.67627661382497510517764685539, 0.974512123645667544407378830245, 1.03737881575641143679977197362, 1.42765051488170276770502346583, 1.70141484667203465306243271462, 1.80637773621327526893608400513, 2.07031885534292481799669935891, 2.25282757288941192961680576392, 2.32548911636716198604692855038, 2.34543124164208192942882904329, 2.82366854276067959334999085121, 2.86106982173926784970592032526, 3.23128852987003543292519806499, 3.43593923365961204052229150795, 3.48164638704211024802711859301, 3.54912240167799957488974557326, 3.64640826041485192048469329655, 3.77865172058802319366260122464, 3.99199358104512779775218746484, 4.13335328543023034725243263650, 4.13591851649453379704390425620, 4.15643175774707877779518431188, 4.72636484814483693695622574735
Plot not available for L-functions of degree greater than 10.
