| L(s) = 1 | − 38·5-s − 243·9-s + 296·11-s − 6.00e3·19-s + 3.74e3·25-s − 1.59e4·29-s + 264·31-s + 1.63e4·41-s + 9.23e3·45-s + 3.65e4·49-s − 1.12e4·55-s + 9.24e4·59-s + 6.25e3·61-s − 1.60e5·71-s + 1.28e5·79-s + 3.93e4·81-s + 7.60e4·89-s + 2.28e5·95-s − 7.19e4·99-s − 1.14e5·101-s − 2.28e5·109-s + 6.04e4·121-s + 3.64e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.679·5-s − 9-s + 0.737·11-s − 3.81·19-s + 1.19·25-s − 3.51·29-s + 0.0493·31-s + 1.51·41-s + 0.679·45-s + 2.17·49-s − 0.501·55-s + 3.45·59-s + 0.215·61-s − 3.78·71-s + 2.32·79-s + 2/3·81-s + 1.01·89-s + 2.59·95-s − 0.737·99-s − 1.11·101-s − 1.83·109-s + 0.375·121-s + 0.0208·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.05516523754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05516523754\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{4} T^{2} )^{3} \) |
| 5 | \( 1 + 38 T - 461 p T^{2} - 9348 p^{2} T^{3} - 461 p^{6} T^{4} + 38 p^{10} T^{5} + p^{15} T^{6} \) |
| good | 7 | \( 1 - 36594 T^{2} + 656281359 T^{4} - 10766281402204 T^{6} + 656281359 p^{10} T^{8} - 36594 p^{20} T^{10} + p^{30} T^{12} \) |
| 11 | \( ( 1 - 148 T + 2621 T^{2} + 31025208 T^{3} + 2621 p^{5} T^{4} - 148 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 13 | \( 1 - 1145958 T^{2} + 753791118135 T^{4} - 322779460890985940 T^{6} + 753791118135 p^{10} T^{8} - 1145958 p^{20} T^{10} + p^{30} T^{12} \) |
| 17 | \( 1 - 2590734 T^{2} + 7157717514399 T^{4} - 10576927881230106084 T^{6} + 7157717514399 p^{10} T^{8} - 2590734 p^{20} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 + 3000 T + 7745097 T^{2} + 14027154000 T^{3} + 7745097 p^{5} T^{4} + 3000 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 - 24763290 T^{2} + 297081932741247 T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + 297081932741247 p^{10} T^{8} - 24763290 p^{20} T^{10} + p^{30} T^{12} \) |
| 29 | \( ( 1 + 7962 T + 79928295 T^{2} + 339128680740 T^{3} + 79928295 p^{5} T^{4} + 7962 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( ( 1 - 132 T + 14583261 T^{2} + 226177330952 T^{3} + 14583261 p^{5} T^{4} - 132 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( 1 - 244387830 T^{2} + 33585500520578247 T^{4} - \)\(28\!\cdots\!40\)\( T^{6} + 33585500520578247 p^{10} T^{8} - 244387830 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( ( 1 - 8170 T + 309202103 T^{2} - 1569581651340 T^{3} + 309202103 p^{5} T^{4} - 8170 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 43 | \( 1 - 45276210 T^{2} + 44529111123627447 T^{4} - \)\(19\!\cdots\!80\)\( T^{6} + 44529111123627447 p^{10} T^{8} - 45276210 p^{20} T^{10} + p^{30} T^{12} \) |
| 47 | \( 1 + 124421046 T^{2} + 29946698142211119 T^{4} + \)\(15\!\cdots\!76\)\( T^{6} + 29946698142211119 p^{10} T^{8} + 124421046 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 - 2370975366 T^{2} + 2397301328123693799 T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + 2397301328123693799 p^{10} T^{8} - 2370975366 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( ( 1 - 46228 T + 2374731725 T^{2} - 58250866741320 T^{3} + 2374731725 p^{5} T^{4} - 46228 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 61 | \( ( 1 - 3126 T + 433417395 T^{2} + 15652173289660 T^{3} + 433417395 p^{5} T^{4} - 3126 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 - 1213193634 T^{2} + 1488268869535589799 T^{4} - \)\(31\!\cdots\!84\)\( T^{6} + 1488268869535589799 p^{10} T^{8} - 1213193634 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( ( 1 + 80400 T + 7314161253 T^{2} + 303984832792800 T^{3} + 7314161253 p^{5} T^{4} + 80400 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 73 | \( 1 - 5305758486 T^{2} + 18815441454973655679 T^{4} - \)\(44\!\cdots\!56\)\( T^{6} + 18815441454973655679 p^{10} T^{8} - 5305758486 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 - 64476 T + 8552089389 T^{2} - 363282494523336 T^{3} + 8552089389 p^{5} T^{4} - 64476 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 12098110050 T^{2} + 79589532006740517447 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + 79589532006740517447 p^{10} T^{8} - 12098110050 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( ( 1 - 38030 T + 13498833047 T^{2} - 295589573171940 T^{3} + 13498833047 p^{5} T^{4} - 38030 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 97 | \( 1 - 7960147590 T^{2} + 45176095155574055247 T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + 45176095155574055247 p^{10} T^{8} - 7960147590 p^{20} T^{10} + p^{30} T^{12} \) |
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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
−7.72019914300101056324490913005, −7.34660930267184447639779965043, −6.99663543264404894854553911034, −6.79660940436603963433598182247, −6.66231629324992382302253734281, −6.36765031417523218956962420017, −6.20981062787582972839024388582, −5.66383613103534572740869211562, −5.64077958887853367515789834866, −5.57423476938506258265881295382, −5.12935293915114756945656540222, −4.70679257414635754484704579280, −4.40739037559595870490914331261, −4.12582078518371921019334462799, −3.91534317820813561339966048519, −3.78917334962657340587010016382, −3.59117184142220902622242379066, −2.78195457348448520337036897411, −2.69696280274648974163018914121, −2.33703431379062242017320316041, −1.92885105946681168061283939816, −1.73461803056784524362756255733, −0.982253561159099034486744916402, −0.59512381136294294449448169094, −0.04559270742026217067887345557,
0.04559270742026217067887345557, 0.59512381136294294449448169094, 0.982253561159099034486744916402, 1.73461803056784524362756255733, 1.92885105946681168061283939816, 2.33703431379062242017320316041, 2.69696280274648974163018914121, 2.78195457348448520337036897411, 3.59117184142220902622242379066, 3.78917334962657340587010016382, 3.91534317820813561339966048519, 4.12582078518371921019334462799, 4.40739037559595870490914331261, 4.70679257414635754484704579280, 5.12935293915114756945656540222, 5.57423476938506258265881295382, 5.64077958887853367515789834866, 5.66383613103534572740869211562, 6.20981062787582972839024388582, 6.36765031417523218956962420017, 6.66231629324992382302253734281, 6.79660940436603963433598182247, 6.99663543264404894854553911034, 7.34660930267184447639779965043, 7.72019914300101056324490913005
Plot not available for L-functions of degree greater than 10.
