| L(s) = 1 | + 7·4-s − 27·9-s + 66·11-s − 31·16-s − 232·19-s − 476·29-s + 184·31-s − 189·36-s − 92·41-s + 462·44-s + 1.48e3·49-s − 2.47e3·59-s + 684·61-s + 375·64-s + 3.63e3·71-s − 1.62e3·76-s + 192·79-s + 486·81-s − 1.67e3·89-s − 1.78e3·99-s + 5.95e3·101-s + 6.87e3·109-s − 3.33e3·116-s + 2.54e3·121-s + 1.28e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 9-s + 1.80·11-s − 0.484·16-s − 2.80·19-s − 3.04·29-s + 1.06·31-s − 7/8·36-s − 0.350·41-s + 1.58·44-s + 4.33·49-s − 5.45·59-s + 1.43·61-s + 0.732·64-s + 6.07·71-s − 2.45·76-s + 0.273·79-s + 2/3·81-s − 1.99·89-s − 1.80·99-s + 5.86·101-s + 6.04·109-s − 2.66·116-s + 1.90·121-s + 0.932·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.545314212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.545314212\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 - p T )^{6} \) |
| good | 2 | \( 1 - 7 T^{2} + 5 p^{4} T^{4} - 9 p^{7} T^{6} + 5 p^{10} T^{8} - 7 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 1486 T^{2} + 1043327 T^{4} - 446281092 T^{6} + 1043327 p^{6} T^{8} - 1486 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 2942 T^{2} + 9743575 T^{4} - 174680996 p^{2} T^{6} + 9743575 p^{6} T^{8} - 2942 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 23094 T^{2} + 242352431 T^{4} - 1503161867252 T^{6} + 242352431 p^{6} T^{8} - 23094 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 + 116 T + 23933 T^{2} + 1591368 T^{3} + 23933 p^{3} T^{4} + 116 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 34858 T^{2} + 545287647 T^{4} - 6518384197324 T^{6} + 545287647 p^{6} T^{8} - 34858 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 238 T + 78303 T^{2} + 11180748 T^{3} + 78303 p^{3} T^{4} + 238 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 - 92 T + 35821 T^{2} + 1288120 T^{3} + 35821 p^{3} T^{4} - 92 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 104418 T^{2} + 8368842375 T^{4} - 531418750549756 T^{6} + 8368842375 p^{6} T^{8} - 104418 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 46 T + 197627 T^{2} + 6094844 T^{3} + 197627 p^{3} T^{4} + 46 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 422950 T^{2} + 78582332263 T^{4} - 8147964327743668 T^{6} + 78582332263 p^{6} T^{8} - 422950 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 305578 T^{2} + 43144732655 T^{4} - 4488744660385164 T^{6} + 43144732655 p^{6} T^{8} - 305578 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 469954 T^{2} + 119328948903 T^{4} - 20999728631062972 T^{6} + 119328948903 p^{6} T^{8} - 469954 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 1236 T + 1040201 T^{2} + 540620792 T^{3} + 1040201 p^{3} T^{4} + 1236 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 - 342 T + 612635 T^{2} - 157910180 T^{3} + 612635 p^{3} T^{4} - 342 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 860786 T^{2} + 488693122807 T^{4} - 169024934493629852 T^{6} + 488693122807 p^{6} T^{8} - 860786 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 1816 T + 2129845 T^{2} - 1498091472 T^{3} + 2129845 p^{3} T^{4} - 1816 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 2176870 T^{2} + 2029742429215 T^{4} - 1038055572700184020 T^{6} + 2029742429215 p^{6} T^{8} - 2176870 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 96 T + 666905 T^{2} + 72496384 T^{3} + 666905 p^{3} T^{4} - 96 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 438390 T^{2} + 263078640471 T^{4} - 94163989842689364 T^{6} + 263078640471 p^{6} T^{8} - 438390 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 838 T + 615287 T^{2} + 488514452 T^{3} + 615287 p^{3} T^{4} + 838 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 2829914 T^{2} + 3973760685391 T^{4} - 3933729961874780972 T^{6} + 3973760685391 p^{6} T^{8} - 2829914 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.09322332178510061089679187157, −4.68722903454362901628490053571, −4.64799749478372910708621657688, −4.38940842471443480560323974266, −4.34405164345601302467216594663, −4.21859974898862005948027952461, −4.11179776452718045006170685983, −3.70537328102698842095350363597, −3.59569481581136268637450347590, −3.26201579935042730506599431038, −3.24578303798602219796874319867, −3.23165999077668277307095940491, −3.18983046544914945612514976341, −2.28453076832656237795547445711, −2.20446374718545994646744891923, −2.16793195719017385793928041165, −2.12017290938942708989069213662, −2.09999517394450833272101786499, −2.05218779474017054605741575589, −1.32895516038687583035557829936, −1.09216001699824977813326038165, −0.984575973238767577934621461867, −0.59301688577943080213199833952, −0.46888439050595172039664651495, −0.20107255527993311670372156453,
0.20107255527993311670372156453, 0.46888439050595172039664651495, 0.59301688577943080213199833952, 0.984575973238767577934621461867, 1.09216001699824977813326038165, 1.32895516038687583035557829936, 2.05218779474017054605741575589, 2.09999517394450833272101786499, 2.12017290938942708989069213662, 2.16793195719017385793928041165, 2.20446374718545994646744891923, 2.28453076832656237795547445711, 3.18983046544914945612514976341, 3.23165999077668277307095940491, 3.24578303798602219796874319867, 3.26201579935042730506599431038, 3.59569481581136268637450347590, 3.70537328102698842095350363597, 4.11179776452718045006170685983, 4.21859974898862005948027952461, 4.34405164345601302467216594663, 4.38940842471443480560323974266, 4.64799749478372910708621657688, 4.68722903454362901628490053571, 5.09322332178510061089679187157
Plot not available for L-functions of degree greater than 10.
