| L(s) = 1 | − 8·4-s − 52·13-s − 69·16-s + 62·19-s − 398·25-s + 82·31-s − 1.13e3·37-s − 1.56e3·43-s + 416·52-s − 886·61-s + 766·64-s − 2.08e3·67-s + 2.39e3·73-s − 496·76-s + 984·79-s + 682·97-s + 3.18e3·100-s + 3.02e3·103-s − 2.47e3·109-s − 1.94e3·121-s − 656·124-s + 127-s + 131-s + 137-s + 139-s + 9.05e3·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 1.10·13-s − 1.07·16-s + 0.748·19-s − 3.18·25-s + 0.475·31-s − 5.02·37-s − 5.55·43-s + 1.10·52-s − 1.85·61-s + 1.49·64-s − 3.80·67-s + 3.84·73-s − 0.748·76-s + 1.40·79-s + 0.713·97-s + 3.18·100-s + 2.89·103-s − 2.17·109-s − 1.46·121-s − 0.475·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.02·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} + 133 T^{4} + 425 p T^{6} + 133 p^{6} T^{8} + p^{15} T^{10} + p^{18} T^{12} \) |
| 5 | \( 1 + 398 T^{2} + 88267 T^{4} + 13515052 T^{6} + 88267 p^{6} T^{8} + 398 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( 1 + 1946 T^{2} + 1291819 T^{4} - 782053340 T^{6} + 1291819 p^{6} T^{8} + 1946 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( ( 1 + 2 p T + 20 p^{2} T^{2} + 150062 T^{3} + 20 p^{5} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 + 19638 T^{2} + 193457043 T^{4} + 1185167951260 T^{6} + 193457043 p^{6} T^{8} + 19638 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 31 T + 11177 T^{2} - 354214 T^{3} + 11177 p^{3} T^{4} - 31 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 + 26874 T^{2} + 137157615 T^{4} - 1070796212084 T^{6} + 137157615 p^{6} T^{8} + 26874 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( 1 + 107486 T^{2} + 5558833147 T^{4} + 171068655050380 T^{6} + 5558833147 p^{6} T^{8} + 107486 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 41 T + 55666 T^{2} - 237883 T^{3} + 55666 p^{3} T^{4} - 41 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( ( 1 + 566 T + 246688 T^{2} + 61584934 T^{3} + 246688 p^{3} T^{4} + 566 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 201446 T^{2} + 24090872899 T^{4} + 1992163622083516 T^{6} + 24090872899 p^{6} T^{8} + 201446 p^{12} T^{10} + p^{18} T^{12} \) |
| 43 | \( ( 1 + 783 T + 433368 T^{2} + 140027659 T^{3} + 433368 p^{3} T^{4} + 783 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 127818 T^{2} + 18065263059 T^{4} + 2679796621113892 T^{6} + 18065263059 p^{6} T^{8} + 127818 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 + 680526 T^{2} + 219567635127 T^{4} + 41561598578306884 T^{6} + 219567635127 p^{6} T^{8} + 680526 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 + 252810 T^{2} + 118459786923 T^{4} + 17779226383468420 T^{6} + 118459786923 p^{6} T^{8} + 252810 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( ( 1 + 443 T + 203324 T^{2} - 18833347 T^{3} + 203324 p^{3} T^{4} + 443 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( ( 1 + 1042 T + 867758 T^{2} + 445253716 T^{3} + 867758 p^{3} T^{4} + 1042 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 71 | \( 1 + 1054578 T^{2} + 491902090755 T^{4} + 2420618974688108 p T^{6} + 491902090755 p^{6} T^{8} + 1054578 p^{12} T^{10} + p^{18} T^{12} \) |
| 73 | \( ( 1 - 1199 T + 1316443 T^{2} - 794269678 T^{3} + 1316443 p^{3} T^{4} - 1199 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 - 492 T + 608946 T^{2} - 546599468 T^{3} + 608946 p^{3} T^{4} - 492 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 834486 T^{2} + 1207789948167 T^{4} - 565854790583996660 T^{6} + 1207789948167 p^{6} T^{8} - 834486 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( 1 + 949814 T^{2} + 691701373123 T^{4} + 216133792389242716 T^{6} + 691701373123 p^{6} T^{8} + 949814 p^{12} T^{10} + p^{18} T^{12} \) |
| 97 | \( ( 1 - 341 T + 490090 T^{2} + 960147395 T^{3} + 490090 p^{3} T^{4} - 341 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.27290667958710477924052552893, −4.88359239990992956705299543976, −4.72868000510255781844456117816, −4.66973403249598545827425700627, −4.63604177812513844000713721965, −4.42080721764398605479863312735, −4.41750350100876189168198675584, −3.78788673680624213131392684620, −3.71854616458668008537437990115, −3.67467351083477055177122819194, −3.50180763645388964696767481365, −3.44453225813179184607177230420, −3.44245629685745685875045094732, −3.13932692664993126265403130134, −2.82807773068587826098911251768, −2.59066875326978552986862920174, −2.33702165415097516486525467925, −2.22537825151505059931539706123, −1.94444790600979433188081345551, −1.92201750575789416896751015963, −1.91419219511509351460931145021, −1.31880122532995929599016682212, −1.28165342310341055228676965163, −1.20937597270489172291112748589, −0.910064134066559161033882164644, 0, 0, 0, 0, 0, 0,
0.910064134066559161033882164644, 1.20937597270489172291112748589, 1.28165342310341055228676965163, 1.31880122532995929599016682212, 1.91419219511509351460931145021, 1.92201750575789416896751015963, 1.94444790600979433188081345551, 2.22537825151505059931539706123, 2.33702165415097516486525467925, 2.59066875326978552986862920174, 2.82807773068587826098911251768, 3.13932692664993126265403130134, 3.44245629685745685875045094732, 3.44453225813179184607177230420, 3.50180763645388964696767481365, 3.67467351083477055177122819194, 3.71854616458668008537437990115, 3.78788673680624213131392684620, 4.41750350100876189168198675584, 4.42080721764398605479863312735, 4.63604177812513844000713721965, 4.66973403249598545827425700627, 4.72868000510255781844456117816, 4.88359239990992956705299543976, 5.27290667958710477924052552893
Plot not available for L-functions of degree greater than 10.
