| L(s) = 1 | + 2·3-s − 4·5-s − 7-s + 9-s − 4·11-s − 4·13-s − 8·15-s + 6·17-s − 2·19-s − 2·21-s + 11·25-s − 4·27-s + 6·29-s + 8·31-s − 8·33-s + 4·35-s + 10·37-s − 8·39-s + 6·41-s − 12·43-s − 4·45-s + 8·47-s + 49-s + 12·51-s − 2·53-s + 16·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s + 11/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 1.39·33-s + 0.676·35-s + 1.64·37-s − 1.28·39-s + 0.937·41-s − 1.82·43-s − 0.596·45-s + 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s + 2.15·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05034187786706, −12.65475575725350, −12.19397481611998, −11.85000687834056, −11.45324661149961, −10.68601684821506, −10.31432042079880, −9.904042238455128, −9.347101296917787, −8.772733729345662, −8.264874941106785, −7.901872590991837, −7.683503912637257, −7.328884057809247, −6.614341754899958, −6.005208617166620, −5.303261125647722, −4.684305184121362, −4.352857596751673, −3.781657526472783, −3.072267010178960, −2.782753404692775, −2.617147184435778, −1.451078606800554, −0.6354340815375321, 0,
0.6354340815375321, 1.451078606800554, 2.617147184435778, 2.782753404692775, 3.072267010178960, 3.781657526472783, 4.352857596751673, 4.684305184121362, 5.303261125647722, 6.005208617166620, 6.614341754899958, 7.328884057809247, 7.683503912637257, 7.901872590991837, 8.264874941106785, 8.772733729345662, 9.347101296917787, 9.904042238455128, 10.31432042079880, 10.68601684821506, 11.45324661149961, 11.85000687834056, 12.19397481611998, 12.65475575725350, 13.05034187786706
