| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s − 4·11-s − 10·13-s − 4·15-s + 4·17-s − 6·19-s + 4·21-s − 4·25-s − 2·27-s − 4·29-s − 8·31-s − 8·33-s − 4·35-s − 4·37-s − 20·39-s + 4·41-s − 4·43-s + 8·47-s + 3·49-s + 8·51-s − 24·53-s + 8·55-s − 12·57-s + 2·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s − 1.20·11-s − 2.77·13-s − 1.03·15-s + 0.970·17-s − 1.37·19-s + 0.872·21-s − 4/5·25-s − 0.384·27-s − 0.742·29-s − 1.43·31-s − 1.39·33-s − 0.676·35-s − 0.657·37-s − 3.20·39-s + 0.624·41-s − 0.609·43-s + 1.16·47-s + 3/7·49-s + 1.12·51-s − 3.29·53-s + 1.07·55-s − 1.58·57-s + 0.260·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.064449560539154830559736415721, −8.592667491135358295514160456258, −8.051020896221726767656579596367, −7.925891354749374689580505613546, −7.74227812686545930648551882902, −7.32516745822968585443267369775, −7.00959202411629396276321208044, −6.35443521257550529302139430751, −5.64306770691211327091872224273, −5.34000762845473403855446184716, −4.93522305709384251515542006625, −4.59223074162802142399922523261, −3.90661891731895904340746586460, −3.68145721929282515553816943297, −2.79696015752342200606741578769, −2.73361712641088580934358634034, −2.11646421928625919274913670418, −1.67498032374400721186743958494, 0, 0,
1.67498032374400721186743958494, 2.11646421928625919274913670418, 2.73361712641088580934358634034, 2.79696015752342200606741578769, 3.68145721929282515553816943297, 3.90661891731895904340746586460, 4.59223074162802142399922523261, 4.93522305709384251515542006625, 5.34000762845473403855446184716, 5.64306770691211327091872224273, 6.35443521257550529302139430751, 7.00959202411629396276321208044, 7.32516745822968585443267369775, 7.74227812686545930648551882902, 7.925891354749374689580505613546, 8.051020896221726767656579596367, 8.592667491135358295514160456258, 9.064449560539154830559736415721
