| L(s) = 1 | + 15·5-s − 8·7-s − 10·11-s + 48·13-s + 37·17-s + 29·19-s − 11·23-s + 150·25-s + 28·29-s + 41·31-s − 120·35-s + 230·37-s + 370·41-s − 130·43-s − 56·47-s − 209·49-s + 805·53-s − 150·55-s + 576·59-s − 257·61-s + 720·65-s − 14·67-s + 1.23e3·71-s − 398·73-s + 80·77-s − 321·79-s + 687·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.431·7-s − 0.274·11-s + 1.02·13-s + 0.527·17-s + 0.350·19-s − 0.0997·23-s + 6/5·25-s + 0.179·29-s + 0.237·31-s − 0.579·35-s + 1.02·37-s + 1.40·41-s − 0.461·43-s − 0.173·47-s − 0.609·49-s + 2.08·53-s − 0.367·55-s + 1.27·59-s − 0.539·61-s + 1.37·65-s − 0.0255·67-s + 2.06·71-s − 0.638·73-s + 0.118·77-s − 0.457·79-s + 0.908·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.725669029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.725669029\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 8 T + 39 p T^{2} + 11320 T^{3} + 39 p^{4} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 1129 T^{2} - 2980 p T^{3} + 1129 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 48 T + 3891 T^{2} - 92328 T^{3} + 3891 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 37 T + 14418 T^{2} - 347965 T^{3} + 14418 p^{3} T^{4} - 37 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 29 T + 17960 T^{2} - 420497 T^{3} + 17960 p^{3} T^{4} - 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 30456 T^{2} + 222899 T^{3} + 30456 p^{3} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 28 T + 55099 T^{2} - 1662544 T^{3} + 55099 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 41 T + 78964 T^{2} - 2734453 T^{3} + 78964 p^{3} T^{4} - 41 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 230 T + 114787 T^{2} - 19298596 T^{3} + 114787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 370 T + 214275 T^{2} - 48796108 T^{3} + 214275 p^{3} T^{4} - 370 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 130 T + 44673 T^{2} + 3936068 T^{3} + 44673 p^{3} T^{4} + 130 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 56 T + 115517 T^{2} + 4195536 T^{3} + 115517 p^{3} T^{4} + 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 805 T + 594702 T^{2} - 247415005 T^{3} + 594702 p^{3} T^{4} - 805 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 576 T + 3639 p T^{2} - 9306072 T^{3} + 3639 p^{4} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 257 T + 202474 T^{2} + 153986701 T^{3} + 202474 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 651149 T^{2} - 22117636 T^{3} + 651149 p^{3} T^{4} + 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1238 T + 1448677 T^{2} - 895305068 T^{3} + 1448677 p^{3} T^{4} - 1238 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 398 T + 1211307 T^{2} + 310732420 T^{3} + 1211307 p^{3} T^{4} + 398 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 321 T + 774636 T^{2} + 459949637 T^{3} + 774636 p^{3} T^{4} + 321 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 687 T + 1037148 T^{2} - 349696983 T^{3} + 1037148 p^{3} T^{4} - 687 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2358 T + 3793395 T^{2} - 3699365060 T^{3} + 3793395 p^{3} T^{4} - 2358 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 576 T + 2295459 T^{2} - 793944704 T^{3} + 2295459 p^{3} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.587010656684076393155624881776, −7.88143878609372245644428302762, −7.86701083377791750343809932117, −7.85720198000421645798473763492, −7.10433232169747824573437845074, −6.95424849503103791492725508232, −6.73052969162843159400036352891, −6.19829944735034210677088141100, −6.08510100228363696984037264424, −6.01465005967817359504035277475, −5.45287213787894197085174093052, −5.26200622227382572126233182091, −5.11349638015423101072384069882, −4.50365074674448642799495439219, −4.20191843680412622711192130156, −3.99832489384659573326825498307, −3.30446223074629078855853957785, −3.23973093975724225646417175892, −2.93698384572121499904008796214, −2.19999523340935722057675932174, −2.18333244276845554111320200577, −1.78580098908290865194078225606, −1.06832719256096467193151386480, −0.75599350679998091551841234850, −0.59891956542668462990713863077,
0.59891956542668462990713863077, 0.75599350679998091551841234850, 1.06832719256096467193151386480, 1.78580098908290865194078225606, 2.18333244276845554111320200577, 2.19999523340935722057675932174, 2.93698384572121499904008796214, 3.23973093975724225646417175892, 3.30446223074629078855853957785, 3.99832489384659573326825498307, 4.20191843680412622711192130156, 4.50365074674448642799495439219, 5.11349638015423101072384069882, 5.26200622227382572126233182091, 5.45287213787894197085174093052, 6.01465005967817359504035277475, 6.08510100228363696984037264424, 6.19829944735034210677088141100, 6.73052969162843159400036352891, 6.95424849503103791492725508232, 7.10433232169747824573437845074, 7.85720198000421645798473763492, 7.86701083377791750343809932117, 7.88143878609372245644428302762, 8.587010656684076393155624881776
