| L(s) = 1 | + 15·5-s + 14·7-s + 22·11-s + 8·13-s + 34·17-s − 4·19-s + 176·23-s + 150·25-s − 98·29-s + 88·31-s + 210·35-s + 284·37-s + 8·41-s + 504·43-s + 280·47-s − 621·49-s + 150·53-s + 330·55-s + 350·59-s + 350·61-s + 120·65-s + 804·67-s + 500·71-s − 486·73-s + 308·77-s + 1.59e3·79-s − 684·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s + 0.603·11-s + 0.170·13-s + 0.485·17-s − 0.0482·19-s + 1.59·23-s + 6/5·25-s − 0.627·29-s + 0.509·31-s + 1.01·35-s + 1.26·37-s + 0.0304·41-s + 1.78·43-s + 0.868·47-s − 1.81·49-s + 0.388·53-s + 0.809·55-s + 0.772·59-s + 0.734·61-s + 0.228·65-s + 1.46·67-s + 0.835·71-s − 0.779·73-s + 0.455·77-s + 2.26·79-s − 0.904·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.15029529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.15029529\) |
| \(L(\frac{5}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - 2 p T + 817 T^{2} - 9196 T^{3} + 817 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 p T + 2149 T^{2} - 69196 T^{3} + 2149 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 2127 T^{2} - 119632 T^{3} + 2127 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 p T + 7951 T^{2} - 430972 T^{3} + 7951 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 7649 T^{2} + 529240 T^{3} + 7649 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 176 T + 35205 T^{2} - 4252064 T^{3} + 35205 p^{3} T^{4} - 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 98 T + 24635 T^{2} + 7058028 T^{3} + 24635 p^{3} T^{4} + 98 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 88 T + 52685 T^{2} - 3535312 T^{3} + 52685 p^{3} T^{4} - 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 284 T + 49559 T^{2} - 13028696 T^{3} + 49559 p^{3} T^{4} - 284 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121451 T^{2} - 10434960 T^{3} + 121451 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 504 T + 174009 T^{2} - 40942288 T^{3} + 174009 p^{3} T^{4} - 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 280 T + 115997 T^{2} - 54406224 T^{3} + 115997 p^{3} T^{4} - 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 150 T + 253683 T^{2} - 44718980 T^{3} + 253683 p^{3} T^{4} - 150 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 350 T + 202005 T^{2} + 17161444 T^{3} + 202005 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 350 T + 204635 T^{2} - 134954132 T^{3} + 204635 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 p T + 1077441 T^{2} - 489158232 T^{3} + 1077441 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 500 T + 1072245 T^{2} - 353953688 T^{3} + 1072245 p^{3} T^{4} - 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 486 T + 1084311 T^{2} + 377044724 T^{3} + 1084311 p^{3} T^{4} + 486 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1592 T + 2161789 T^{2} - 1645964560 T^{3} + 2161789 p^{3} T^{4} - 1592 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 684 T + 1381713 T^{2} + 561532168 T^{3} + 1381713 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 668 T + 2135307 T^{2} - 937072184 T^{3} + 2135307 p^{3} T^{4} - 668 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1394 T + 2868623 T^{2} + 2439375164 T^{3} + 2868623 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.327354213251203685812491872786, −7.66989436275914303851749250651, −7.56927378934543411240532753381, −7.40572822470370450200737239686, −6.73298807393961043426792631396, −6.73060017982676221252337766656, −6.65226096481712868977592850336, −5.89073612469584847112310058556, −5.84036727135581625349582303971, −5.83775254906170227105516882193, −5.11527604403730804942307575895, −5.00067207131618732558532909073, −4.97389523542831345951528206394, −4.21224768658033662536849521249, −4.15528441469696365237606685948, −3.87034662897963979607725059497, −3.18621961135979935708094206377, −2.99808152661762629957099906458, −2.78005104036952292404722075943, −2.15220079774395696503825157241, −1.84756298653822527608429440287, −1.79244550709812595813468615436, −0.966778151300225403399521368991, −0.818345420903198442924507445162, −0.64710079732116236320918425889,
0.64710079732116236320918425889, 0.818345420903198442924507445162, 0.966778151300225403399521368991, 1.79244550709812595813468615436, 1.84756298653822527608429440287, 2.15220079774395696503825157241, 2.78005104036952292404722075943, 2.99808152661762629957099906458, 3.18621961135979935708094206377, 3.87034662897963979607725059497, 4.15528441469696365237606685948, 4.21224768658033662536849521249, 4.97389523542831345951528206394, 5.00067207131618732558532909073, 5.11527604403730804942307575895, 5.83775254906170227105516882193, 5.84036727135581625349582303971, 5.89073612469584847112310058556, 6.65226096481712868977592850336, 6.73060017982676221252337766656, 6.73298807393961043426792631396, 7.40572822470370450200737239686, 7.56927378934543411240532753381, 7.66989436275914303851749250651, 8.327354213251203685812491872786
