| L(s) = 1 | + 7·4-s − 729·9-s − 4.04e3·16-s − 2.63e4·19-s − 1.15e4·31-s − 5.10e3·36-s − 2.35e5·49-s − 6.51e5·61-s − 5.70e4·64-s − 1.84e5·76-s + 1.78e6·79-s + 5.31e5·81-s − 1.54e6·109-s + 3.54e6·121-s − 8.06e4·124-s + 127-s + 131-s + 137-s + 139-s + 2.95e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.65e6·169-s + 1.92e7·171-s + ⋯ |
| L(s) = 1 | + 7/64·4-s − 9-s − 0.988·16-s − 3.84·19-s − 0.386·31-s − 0.109·36-s − 2·49-s − 2.87·61-s − 0.217·64-s − 0.420·76-s + 3.62·79-s + 81-s − 1.19·109-s + 2·121-s − 0.0422·124-s + 0.988·144-s − 2·169-s + 3.84·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3379941154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3379941154\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 39972098 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 13178 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 81332062 T^{2} + p^{12} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5758 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13452309502 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 36467062702 T^{2} + p^{12} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 325798 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 893662 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 433407300622 T^{2} + p^{12} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37164285903808327222601417287, −13.25246173704196760060345612422, −12.31727234369541275557580909901, −12.19559694379489734808109894813, −11.06103247038808431914848051559, −10.97262444962791215726257107593, −10.53900315230269861718560469397, −9.491340265099824635365906148961, −9.037494474382992978939835329412, −8.366816100079059101435449016318, −8.085887142896646920623401070473, −7.01142755089926863121332520628, −6.24484916790120163634360117631, −6.16539567150352980045317031992, −4.87327394538922342878671774047, −4.40658762032263635968638622306, −3.44715996632572020628399193640, −2.39837308512210718856618950413, −1.88028761035958116542280760197, −0.20611837401749712340404433988,
0.20611837401749712340404433988, 1.88028761035958116542280760197, 2.39837308512210718856618950413, 3.44715996632572020628399193640, 4.40658762032263635968638622306, 4.87327394538922342878671774047, 6.16539567150352980045317031992, 6.24484916790120163634360117631, 7.01142755089926863121332520628, 8.085887142896646920623401070473, 8.366816100079059101435449016318, 9.037494474382992978939835329412, 9.491340265099824635365906148961, 10.53900315230269861718560469397, 10.97262444962791215726257107593, 11.06103247038808431914848051559, 12.19559694379489734808109894813, 12.31727234369541275557580909901, 13.25246173704196760060345612422, 13.37164285903808327222601417287
