| L(s) = 1 | − 18·7-s − 18·13-s − 42·19-s + 60·25-s + 18·31-s + 78·37-s − 78·43-s + 63·49-s + 96·61-s + 12·67-s + 96·73-s − 54·79-s + 324·91-s + 390·97-s + 120·103-s + 378·109-s + 84·121-s + 127-s + 131-s + 756·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.57·7-s − 1.38·13-s − 2.21·19-s + 12/5·25-s + 0.580·31-s + 2.10·37-s − 1.81·43-s + 9/7·49-s + 1.57·61-s + 0.179·67-s + 1.31·73-s − 0.683·79-s + 3.56·91-s + 4.02·97-s + 1.16·103-s + 3.46·109-s + 0.694·121-s + 0.00787·127-s + 0.00763·131-s + 5.68·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.211805570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211805570\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12 p T^{2} + 1719 T^{4} - 42168 T^{6} + 1719 p^{4} T^{8} - 12 p^{9} T^{10} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 9 T + 90 T^{2} + 801 T^{3} + 90 p^{2} T^{4} + 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( 1 - 84 T^{2} + 22359 T^{4} - 1676904 T^{6} + 22359 p^{4} T^{8} - 84 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 + 9 T + 6 p T^{2} - 2011 T^{3} + 6 p^{3} T^{4} + 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 1254 T^{2} + 727839 T^{4} - 259019124 T^{6} + 727839 p^{4} T^{8} - 1254 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 + 21 T + 894 T^{2} + 14305 T^{3} + 894 p^{2} T^{4} + 21 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 1542 T^{2} + 1527615 T^{4} - 943028916 T^{6} + 1527615 p^{4} T^{8} - 1542 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 4260 T^{2} + 8039991 T^{4} - 8685384392 T^{6} + 8039991 p^{4} T^{8} - 4260 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 - 9 T + 2454 T^{2} - 395 p T^{3} + 2454 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - 39 T + 3726 T^{2} - 100059 T^{3} + 3726 p^{2} T^{4} - 39 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 - 7068 T^{2} + 24108807 T^{4} - 50220858872 T^{6} + 24108807 p^{4} T^{8} - 7068 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 + 39 T + 4986 T^{2} + 119111 T^{3} + 4986 p^{2} T^{4} + 39 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 5556 T^{2} + 23961351 T^{4} - 59149102376 T^{6} + 23961351 p^{4} T^{8} - 5556 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 10452 T^{2} + 49640535 T^{4} - 157869238632 T^{6} + 49640535 p^{4} T^{8} - 10452 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 6828 T^{2} + 51192183 T^{4} - 175546291800 T^{6} + 51192183 p^{4} T^{8} - 6828 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 48 T + 4719 T^{2} - 457168 T^{3} + 4719 p^{2} T^{4} - 48 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 - 6 T + 12135 T^{2} - 41972 T^{3} + 12135 p^{2} T^{4} - 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 22164 T^{2} + 236630631 T^{4} - 1507046917224 T^{6} + 236630631 p^{4} T^{8} - 22164 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 48 T + 12327 T^{2} - 463408 T^{3} + 12327 p^{2} T^{4} - 48 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 + 27 T + 6990 T^{2} + 268871 T^{3} + 6990 p^{2} T^{4} + 27 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( 1 - 27798 T^{2} + 366680655 T^{4} - 3066353533524 T^{6} + 366680655 p^{4} T^{8} - 27798 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 + 1812 T^{2} + 126027015 T^{4} + 344218999336 T^{6} + 126027015 p^{4} T^{8} + 1812 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 - 195 T + 24474 T^{2} - 2173451 T^{3} + 24474 p^{2} T^{4} - 195 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−4.56006323324513255254019006199, −4.32571237711864051479469908210, −4.31568447170200737705099796830, −4.21372522880070769862078818538, −4.09262918280045396435009570888, −3.65770488637991377017931960711, −3.50385398179810296161195148006, −3.47791053686829426667964645201, −3.36942797140730680268929926166, −3.16300628887806108429041999746, −3.15694704960666279028197380251, −2.76512594513296234159293564464, −2.64421739858444495291910326591, −2.47348285477547855236572720110, −2.38649794360092315794144146125, −2.18241859406386602942513323829, −2.15855489635934425390904030734, −1.71960280213355898383109310223, −1.58795059941997347379876726772, −1.20779248762583632403671677368, −1.11919057043918080911408917018, −0.64780519200320730425673331939, −0.49018937451990268167362850987, −0.48425510424418085637467801057, −0.18975158733192574020961281072,
0.18975158733192574020961281072, 0.48425510424418085637467801057, 0.49018937451990268167362850987, 0.64780519200320730425673331939, 1.11919057043918080911408917018, 1.20779248762583632403671677368, 1.58795059941997347379876726772, 1.71960280213355898383109310223, 2.15855489635934425390904030734, 2.18241859406386602942513323829, 2.38649794360092315794144146125, 2.47348285477547855236572720110, 2.64421739858444495291910326591, 2.76512594513296234159293564464, 3.15694704960666279028197380251, 3.16300628887806108429041999746, 3.36942797140730680268929926166, 3.47791053686829426667964645201, 3.50385398179810296161195148006, 3.65770488637991377017931960711, 4.09262918280045396435009570888, 4.21372522880070769862078818538, 4.31568447170200737705099796830, 4.32571237711864051479469908210, 4.56006323324513255254019006199
Plot not available for L-functions of degree greater than 10.
