| L(s) = 1 | + 9·3-s − 6·5-s + 6·7-s + 36·9-s + 6·11-s − 15·13-s − 54·15-s − 9·17-s + 18·19-s + 54·21-s + 18·23-s + 18·25-s + 80·27-s − 3·29-s − 3·31-s + 54·33-s − 36·35-s − 24·37-s − 135·39-s − 15·41-s − 21·43-s − 216·45-s + 21·47-s + 30·49-s − 81·51-s − 27·53-s − 36·55-s + ⋯ |
| L(s) = 1 | + 5.19·3-s − 2.68·5-s + 2.26·7-s + 12·9-s + 1.80·11-s − 4.16·13-s − 13.9·15-s − 2.18·17-s + 4.12·19-s + 11.7·21-s + 3.75·23-s + 18/5·25-s + 15.3·27-s − 0.557·29-s − 0.538·31-s + 9.40·33-s − 6.08·35-s − 3.94·37-s − 21.6·39-s − 2.34·41-s − 3.20·43-s − 32.1·45-s + 3.06·47-s + 30/7·49-s − 11.3·51-s − 3.70·53-s − 4.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.798805547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.798805547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 18 T + 144 T^{2} - 737 T^{3} + 144 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - p^{2} T + 5 p^{2} T^{2} - 161 T^{3} + 50 p^{2} T^{4} - 38 p^{3} T^{5} + 1945 T^{6} - 38 p^{4} T^{7} + 50 p^{4} T^{8} - 161 p^{3} T^{9} + 5 p^{6} T^{10} - p^{7} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 6 T + 6 T^{2} - 6 T^{3} + 24 p T^{4} - 384 T^{5} + 191 T^{6} - 384 p T^{7} + 24 p^{3} T^{8} - 6 p^{3} T^{9} + 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T - 6 T^{2} + 14 T^{3} + 504 T^{4} - 768 T^{5} - 3409 T^{6} - 768 p T^{7} + 504 p^{2} T^{8} + 14 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 15 T + 96 T^{2} + 298 T^{3} - 9 p T^{4} - 6255 T^{5} - 32679 T^{6} - 6255 p T^{7} - 9 p^{3} T^{8} + 298 p^{3} T^{9} + 96 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 9 T + 36 T^{2} + 40 T^{3} - 369 T^{4} - 2673 T^{5} - 10927 T^{6} - 2673 p T^{7} - 369 p^{2} T^{8} + 40 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 18 T + 180 T^{2} - 1354 T^{3} + 8784 T^{4} - 49464 T^{5} + 249209 T^{6} - 49464 p T^{7} + 8784 p^{2} T^{8} - 1354 p^{3} T^{9} + 180 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 9921 p T^{7} + 1872 p^{2} T^{8} + 378 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 3 T - 60 T^{2} - 59 T^{3} + 2223 T^{4} - 774 T^{5} - 78969 T^{6} - 774 p T^{7} + 2223 p^{2} T^{8} - 59 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 12 T + 150 T^{2} + 907 T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 15 T + 150 T^{2} + 1000 T^{3} + 8025 T^{4} + 61965 T^{5} + 472529 T^{6} + 61965 p T^{7} + 8025 p^{2} T^{8} + 1000 p^{3} T^{9} + 150 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 21 T + 174 T^{2} + 454 T^{3} - 3879 T^{4} - 58707 T^{5} - 460263 T^{6} - 58707 p T^{7} - 3879 p^{2} T^{8} + 454 p^{3} T^{9} + 174 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 21 T + 186 T^{2} - 908 T^{3} + 2715 T^{4} - 657 T^{5} - 54025 T^{6} - 657 p T^{7} + 2715 p^{2} T^{8} - 908 p^{3} T^{9} + 186 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 27 T + 324 T^{2} + 2178 T^{3} + 11259 T^{4} + 87453 T^{5} + 753949 T^{6} + 87453 p T^{7} + 11259 p^{2} T^{8} + 2178 p^{3} T^{9} + 324 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 6 T + 6 T^{2} - 332 T^{3} - 2208 T^{4} + 10908 T^{5} + 340589 T^{6} + 10908 p T^{7} - 2208 p^{2} T^{8} - 332 p^{3} T^{9} + 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 12 T - 24 T^{2} + 1238 T^{3} - 7956 T^{4} - 48996 T^{5} + 978639 T^{6} - 48996 p T^{7} - 7956 p^{2} T^{8} + 1238 p^{3} T^{9} - 24 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 18 T + 252 T^{2} - 1838 T^{3} + 4968 T^{4} + 64692 T^{5} - 990987 T^{6} + 64692 p T^{7} + 4968 p^{2} T^{8} - 1838 p^{3} T^{9} + 252 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 252 T^{2} - 1814 T^{3} + 5832 T^{4} + 54324 T^{5} - 887815 T^{6} + 54324 p T^{7} + 5832 p^{2} T^{8} - 1814 p^{3} T^{9} + 252 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 36 T + 576 T^{2} - 5760 T^{3} + 51984 T^{4} - 544320 T^{5} + 5272343 T^{6} - 544320 p T^{7} + 51984 p^{2} T^{8} - 5760 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 27 T + 180 T^{2} + 1964 T^{3} - 32490 T^{4} + 11547 T^{5} + 1937037 T^{6} + 11547 p T^{7} - 32490 p^{2} T^{8} + 1964 p^{3} T^{9} + 180 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 12 T - 150 T^{2} + 562 T^{3} + 37494 T^{4} - 98250 T^{5} - 2840605 T^{6} - 98250 p T^{7} + 37494 p^{2} T^{8} + 562 p^{3} T^{9} - 150 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 69891 p T^{7} - 279 p^{2} T^{8} + 1035 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 18 T + 270 T^{2} + 4178 T^{3} + 50940 T^{4} + 535752 T^{5} + 5820555 T^{6} + 535752 p T^{7} + 50940 p^{2} T^{8} + 4178 p^{3} T^{9} + 270 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−7.51076243968638687831261716307, −7.19023234549246917322331940599, −7.10304176251575301401510344830, −6.86760311265819032156847237331, −6.86506098787827665692427074576, −6.81095971374930485971199814768, −6.36941495451777279764713634337, −5.17734418263665096148452078680, −5.15016315537300656932509088054, −5.13811659221697972602746431599, −5.12223883633110212299795092473, −4.97371770631896980555020523649, −4.91553338292048600545727136431, −4.11487083630316234365699608302, −3.89247652386135324755501241241, −3.86470832112916888284214346073, −3.45479081988621444808383464768, −3.34621013475549004826138704423, −3.33461707007631781166032241626, −2.92833934409841996085641645278, −2.64611949732025841826463406682, −2.31746176230078828079597630110, −2.30381900891947702958730734205, −1.79194421800063348002325011051, −1.27680565541480310475977999137,
1.27680565541480310475977999137, 1.79194421800063348002325011051, 2.30381900891947702958730734205, 2.31746176230078828079597630110, 2.64611949732025841826463406682, 2.92833934409841996085641645278, 3.33461707007631781166032241626, 3.34621013475549004826138704423, 3.45479081988621444808383464768, 3.86470832112916888284214346073, 3.89247652386135324755501241241, 4.11487083630316234365699608302, 4.91553338292048600545727136431, 4.97371770631896980555020523649, 5.12223883633110212299795092473, 5.13811659221697972602746431599, 5.15016315537300656932509088054, 5.17734418263665096148452078680, 6.36941495451777279764713634337, 6.81095971374930485971199814768, 6.86506098787827665692427074576, 6.86760311265819032156847237331, 7.10304176251575301401510344830, 7.19023234549246917322331940599, 7.51076243968638687831261716307
Plot not available for L-functions of degree greater than 10.
