L(s) = 1 | + 4·3-s + 7·9-s − 4·11-s − 4·13-s − 2·17-s − 4·19-s − 8·23-s − 5·25-s + 4·27-s − 16·33-s − 16·39-s + 8·47-s − 5·49-s − 8·51-s − 16·57-s + 8·67-s − 32·69-s − 12·73-s − 20·75-s + 8·79-s − 8·81-s + 14·89-s − 28·99-s − 16·103-s − 28·117-s + 5·121-s + 127-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 7/3·9-s − 1.20·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 25-s + 0.769·27-s − 2.78·33-s − 2.56·39-s + 1.16·47-s − 5/7·49-s − 1.12·51-s − 2.11·57-s + 0.977·67-s − 3.85·69-s − 1.40·73-s − 2.30·75-s + 0.900·79-s − 8/9·81-s + 1.48·89-s − 2.81·99-s − 1.57·103-s − 2.58·117-s + 5/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
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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
−8.356367122239688119465632523598, −8.013161623003913810526172289868, −7.75083754905669529596832792615, −7.41923409947571712417361584709, −6.73374282118455424420870009136, −6.15593493399353886403929655220, −5.55558341586249995711814231944, −4.99375321774290804247339921270, −4.21612644919682573004336207208, −3.99278789798550888415055709377, −3.22791246775205293569541102966, −2.72124716201087153697626339042, −2.13796367393343059861647482431, −2.02864381350513604826483179141, 0,
2.02864381350513604826483179141, 2.13796367393343059861647482431, 2.72124716201087153697626339042, 3.22791246775205293569541102966, 3.99278789798550888415055709377, 4.21612644919682573004336207208, 4.99375321774290804247339921270, 5.55558341586249995711814231944, 6.15593493399353886403929655220, 6.73374282118455424420870009136, 7.41923409947571712417361584709, 7.75083754905669529596832792615, 8.013161623003913810526172289868, 8.356367122239688119465632523598