L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s + 8·11-s − 4·13-s + 8·15-s + 4·17-s − 8·19-s − 16·23-s + 2·25-s − 4·27-s + 12·29-s + 16·31-s − 16·33-s + 12·37-s + 8·39-s − 12·41-s + 8·43-s − 12·45-s − 14·49-s − 8·51-s − 4·53-s − 32·55-s + 16·57-s + 8·59-s − 4·61-s + 16·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 2.41·11-s − 1.10·13-s + 2.06·15-s + 0.970·17-s − 1.83·19-s − 3.33·23-s + 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 2.78·33-s + 1.97·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 1.78·45-s − 2·49-s − 1.12·51-s − 0.549·53-s − 4.31·55-s + 2.11·57-s + 1.04·59-s − 0.512·61-s + 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2906599836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2906599836\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−19.8934316645, −19.8934316645, −19.1551537949, −19.1551537949, −17.6707519500, −17.6707519500, −16.6272610009, −16.6272610009, −15.4732516475, −15.4732516475, −14.1819882616, −14.1819882616, −12.3172568607, −12.3172568607, −11.5820474614, −11.5820474614, −9.96467191573, −9.96467191573, −8.09899069409, −8.09899069409, −6.42897107073, −6.42897107073, −4.25303028693, −4.25303028693,
4.25303028693, 4.25303028693, 6.42897107073, 6.42897107073, 8.09899069409, 8.09899069409, 9.96467191573, 9.96467191573, 11.5820474614, 11.5820474614, 12.3172568607, 12.3172568607, 14.1819882616, 14.1819882616, 15.4732516475, 15.4732516475, 16.6272610009, 16.6272610009, 17.6707519500, 17.6707519500, 19.1551537949, 19.1551537949, 19.8934316645, 19.8934316645