| L(s) = 1 | − 1.65·2-s − 2.71·3-s + 0.731·4-s + 1.07·5-s + 4.48·6-s + 0.230·7-s + 2.09·8-s + 4.37·9-s − 1.78·10-s + 3.46·11-s − 1.98·12-s − 3.89·13-s − 0.381·14-s − 2.92·15-s − 4.92·16-s + 2.48·17-s − 7.22·18-s − 3.19·19-s + 0.789·20-s − 0.626·21-s − 5.72·22-s + 6.87·23-s − 5.69·24-s − 3.83·25-s + 6.43·26-s − 3.72·27-s + 0.168·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 1.56·3-s + 0.365·4-s + 0.482·5-s + 1.83·6-s + 0.0872·7-s + 0.741·8-s + 1.45·9-s − 0.563·10-s + 1.04·11-s − 0.573·12-s − 1.08·13-s − 0.101·14-s − 0.755·15-s − 1.23·16-s + 0.603·17-s − 1.70·18-s − 0.732·19-s + 0.176·20-s − 0.136·21-s − 1.22·22-s + 1.43·23-s − 1.16·24-s − 0.767·25-s + 1.26·26-s − 0.717·27-s + 0.0319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 - 0.230T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 0.351T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
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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
−8.629838361673644956381015594637, −7.63653715950667394180137408972, −6.90103680189507749475192781884, −6.39259754426152209142789260238, −5.29822362314210698730326939495, −4.88907870685453479296521561176, −3.77690055512750344712746318378, −2.03509638828172788185918418016, −1.11849377325556836078590045339, 0,
1.11849377325556836078590045339, 2.03509638828172788185918418016, 3.77690055512750344712746318378, 4.88907870685453479296521561176, 5.29822362314210698730326939495, 6.39259754426152209142789260238, 6.90103680189507749475192781884, 7.63653715950667394180137408972, 8.629838361673644956381015594637
