| 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
| \(L(2)\) |
\(\approx\) |
\(3.210979868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.210979868\) |
| \(L(\frac{5}{2})\) |
|
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
−8.156923084275259142357132326469, −7.84348810453179691952009224403, −7.76386217967641117470558250795, −7.44374995651393907735832396752, −6.94391772419545452099959240841, −6.76414390490200193824605851762, −6.54035472937104910806124221451, −6.14821797196178524810314748528, −5.76776152804643896076762940241, −5.76362286617551410058986176010, −5.01839642686629740296460525242, −5.01033677885088299644020045160, −4.62594410266643049906489962078, −4.31196834111572899981380929453, −4.04710530666978098793405007868, −3.98661240473918553423636050978, −3.23420983492646690310836600826, −3.01360899133235319223619934602, −2.97785393838528215489297049865, −2.14325014398072126502342668984, −1.99172689691768488254924794949, −1.71280162844390892367324909853, −0.910212105004146185273631306175, −0.61340604972973596111208984606, −0.38856088489399866808858452219,
0.38856088489399866808858452219, 0.61340604972973596111208984606, 0.910212105004146185273631306175, 1.71280162844390892367324909853, 1.99172689691768488254924794949, 2.14325014398072126502342668984, 2.97785393838528215489297049865, 3.01360899133235319223619934602, 3.23420983492646690310836600826, 3.98661240473918553423636050978, 4.04710530666978098793405007868, 4.31196834111572899981380929453, 4.62594410266643049906489962078, 5.01033677885088299644020045160, 5.01839642686629740296460525242, 5.76362286617551410058986176010, 5.76776152804643896076762940241, 6.14821797196178524810314748528, 6.54035472937104910806124221451, 6.76414390490200193824605851762, 6.94391772419545452099959240841, 7.44374995651393907735832396752, 7.76386217967641117470558250795, 7.84348810453179691952009224403, 8.156923084275259142357132326469
