| L(s) = 1 | + 2-s − 4·3-s + 4-s + 8·5-s − 4·6-s − 22·7-s − 15·8-s − 3·9-s + 8·10-s + 28·11-s − 4·12-s + 30·13-s − 22·14-s − 32·15-s − 31·16-s − 3·18-s + 80·19-s + 8·20-s + 88·21-s + 28·22-s − 142·23-s + 60·24-s − 267·25-s + 30·26-s + 76·27-s − 22·28-s + 456·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.769·3-s + 1/8·4-s + 0.715·5-s − 0.272·6-s − 1.18·7-s − 0.662·8-s − 1/9·9-s + 0.252·10-s + 0.767·11-s − 0.0962·12-s + 0.640·13-s − 0.419·14-s − 0.550·15-s − 0.484·16-s − 0.0392·18-s + 0.965·19-s + 0.0894·20-s + 0.914·21-s + 0.271·22-s − 1.28·23-s + 0.510·24-s − 2.13·25-s + 0.226·26-s + 0.541·27-s − 0.148·28-s + 2.91·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24137569 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24137569 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.725054675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725054675\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p^{4} T^{3} - p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + 19 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 8 T + 331 T^{2} - 2032 T^{3} + 331 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 22 T + 891 T^{2} + 14300 T^{3} + 891 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 2627 T^{2} - 69844 T^{3} + 2627 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 30 T + 5119 T^{2} - 141212 T^{3} + 5119 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 80 T + 15945 T^{2} - 757312 T^{3} + 15945 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 142 T + 20731 T^{2} + 1854884 T^{3} + 20731 p^{3} T^{4} + 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 456 T + 127075 T^{2} - 23761392 T^{3} + 127075 p^{3} T^{4} - 456 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 230 T + 77787 T^{2} + 13785468 T^{3} + 77787 p^{3} T^{4} + 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 356 T + 133995 T^{2} + 29888184 T^{3} + 133995 p^{3} T^{4} + 356 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 294 T + 120199 T^{2} - 38886804 T^{3} + 120199 p^{3} T^{4} - 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 556 T + 289617 T^{2} - 81141512 T^{3} + 289617 p^{3} T^{4} - 556 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 640 T + 396797 T^{2} - 134564608 T^{3} + 396797 p^{3} T^{4} - 640 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 302 T + 293171 T^{2} - 71759636 T^{3} + 293171 p^{3} T^{4} - 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 636 T + 514369 T^{2} - 211823016 T^{3} + 514369 p^{3} T^{4} - 636 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 84 T + 556531 T^{2} - 31340024 T^{3} + 556531 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1008 T + 967329 T^{2} - 607104160 T^{3} + 967329 p^{3} T^{4} - 1008 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 402 T + 483859 T^{2} - 12894428 T^{3} + 483859 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 838 T + 1394903 T^{2} + 671950004 T^{3} + 1394903 p^{3} T^{4} + 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 594 T + 357843 T^{2} + 156405492 T^{3} + 357843 p^{3} T^{4} - 594 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2396 T + 3204249 T^{2} + 2882084008 T^{3} + 3204249 p^{3} T^{4} + 2396 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 170 T + 1042603 T^{2} - 206881916 T^{3} + 1042603 p^{3} T^{4} + 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 270 T + 2151919 T^{2} - 286220420 T^{3} + 2151919 p^{3} T^{4} - 270 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16152795725608277697571542451, −9.700804363694586511544341549430, −9.645152115890941962420273654958, −9.376462070625949273247621168080, −8.646066188835189082465595813330, −8.558900681116993448199272783299, −8.531408892990994087616245405627, −7.54007051531059185708868274122, −7.46273662908480002917341562695, −7.06120151071962756761984013898, −6.50489073890688111542814836182, −6.29819334456106865558532932530, −5.95133000359171548499051263504, −5.92891169100891324104834455317, −5.39385682582968923204369289712, −5.20679053668716813955100439581, −4.25134514507820346701216094376, −4.18155433044405949340996349645, −3.76790985156087331662988422445, −3.03319153829326505427247663103, −2.99353576388426351759362661057, −2.14031467337239665624824831578, −1.81519181062734399998270758839, −0.76398739206330964468500475345, −0.54061341279922133474278440778,
0.54061341279922133474278440778, 0.76398739206330964468500475345, 1.81519181062734399998270758839, 2.14031467337239665624824831578, 2.99353576388426351759362661057, 3.03319153829326505427247663103, 3.76790985156087331662988422445, 4.18155433044405949340996349645, 4.25134514507820346701216094376, 5.20679053668716813955100439581, 5.39385682582968923204369289712, 5.92891169100891324104834455317, 5.95133000359171548499051263504, 6.29819334456106865558532932530, 6.50489073890688111542814836182, 7.06120151071962756761984013898, 7.46273662908480002917341562695, 7.54007051531059185708868274122, 8.531408892990994087616245405627, 8.558900681116993448199272783299, 8.646066188835189082465595813330, 9.376462070625949273247621168080, 9.645152115890941962420273654958, 9.700804363694586511544341549430, 10.16152795725608277697571542451
