| L(s) = 1 | + 3·3-s − 4·4-s − 13·7-s − 12·12-s + 46·13-s − 11·19-s − 39·21-s − 25·25-s − 27·27-s + 52·28-s + 13·31-s + 73·37-s + 138·39-s − 122·43-s + 120·49-s − 184·52-s − 33·57-s − 74·61-s + 64·64-s + 13·67-s + 97·73-s − 75·75-s + 44·76-s − 11·79-s − 81·81-s + 156·84-s − 598·91-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s − 1.85·7-s − 12-s + 3.53·13-s − 0.578·19-s − 1.85·21-s − 25-s − 27-s + 13/7·28-s + 0.419·31-s + 1.97·37-s + 3.53·39-s − 2.83·43-s + 2.44·49-s − 3.53·52-s − 0.578·57-s − 1.21·61-s + 64-s + 0.194·67-s + 1.32·73-s − 75-s + 0.578·76-s − 0.139·79-s − 81-s + 13/7·84-s − 6.57·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7789239932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7789239932\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 13 T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 - 26 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−18.22591168197302180246445512932, −18.18469278944139048887368297192, −16.84208441965893813213973103800, −16.41480989137551148618680286450, −15.53367582927521313680246387256, −15.42721410848876837977762970691, −14.19262124482623320311386052803, −13.54581719826265743074080423457, −13.25824231990411862652613133590, −13.05608419011204828296859916744, −11.67230751947332469346862091920, −10.85867357147380919384377244288, −9.853380481351000633855114089440, −9.252588891088115612641962602212, −8.616907772579954865039179977931, −8.157996158947256526575565289898, −6.45358831043530219069890543980, −6.00577066649538924854715699344, −3.95940027798559613597211892625, −3.36578704399175758378478904088,
3.36578704399175758378478904088, 3.95940027798559613597211892625, 6.00577066649538924854715699344, 6.45358831043530219069890543980, 8.157996158947256526575565289898, 8.616907772579954865039179977931, 9.252588891088115612641962602212, 9.853380481351000633855114089440, 10.85867357147380919384377244288, 11.67230751947332469346862091920, 13.05608419011204828296859916744, 13.25824231990411862652613133590, 13.54581719826265743074080423457, 14.19262124482623320311386052803, 15.42721410848876837977762970691, 15.53367582927521313680246387256, 16.41480989137551148618680286450, 16.84208441965893813213973103800, 18.18469278944139048887368297192, 18.22591168197302180246445512932
