| L(s) = 1 | + 2-s + 0.517·3-s + 4-s + 5-s + 0.517·6-s − 2.93·7-s + 8-s − 2.73·9-s + 10-s − 3.03·11-s + 0.517·12-s + 6.00·13-s − 2.93·14-s + 0.517·15-s + 16-s − 3.44·17-s − 2.73·18-s + 2.66·19-s + 20-s − 1.51·21-s − 3.03·22-s + 0.517·24-s + 25-s + 6.00·26-s − 2.96·27-s − 2.93·28-s + 1.13·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.298·3-s + 0.5·4-s + 0.447·5-s + 0.211·6-s − 1.10·7-s + 0.353·8-s − 0.910·9-s + 0.316·10-s − 0.915·11-s + 0.149·12-s + 1.66·13-s − 0.783·14-s + 0.133·15-s + 0.250·16-s − 0.836·17-s − 0.643·18-s + 0.611·19-s + 0.223·20-s − 0.331·21-s − 0.647·22-s + 0.105·24-s + 0.200·25-s + 1.17·26-s − 0.571·27-s − 0.554·28-s + 0.211·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.517T + 3T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 6.00T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 9.53T + 37T^{2} \) |
| 41 | \( 1 + 0.303T + 41T^{2} \) |
| 43 | \( 1 + 0.929T + 43T^{2} \) |
| 47 | \( 1 + 6.06T + 47T^{2} \) |
| 53 | \( 1 + 0.578T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 5.72T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.79672506194456406274460688188, −6.88690546030701028411018420602, −6.28297749732419558391113739284, −5.67227301186934987046185902521, −5.09783788891599865162521642705, −3.86690726948082096669923954143, −3.28597379251596445670329308946, −2.66668628616524023896957716406, −1.62647960434481503447499914786, 0,
1.62647960434481503447499914786, 2.66668628616524023896957716406, 3.28597379251596445670329308946, 3.86690726948082096669923954143, 5.09783788891599865162521642705, 5.67227301186934987046185902521, 6.28297749732419558391113739284, 6.88690546030701028411018420602, 7.79672506194456406274460688188
