| L(s) = 1 | − 2.55·2-s + 2.61·3-s + 4.50·4-s − 3.64·5-s − 6.66·6-s + 1.17·7-s − 6.40·8-s + 3.83·9-s + 9.28·10-s − 4.29·11-s + 11.7·12-s − 2.96·13-s − 2.99·14-s − 9.51·15-s + 7.31·16-s + 1.01·17-s − 9.77·18-s + 8.16·19-s − 16.4·20-s + 3.07·21-s + 10.9·22-s − 1.14·23-s − 16.7·24-s + 8.25·25-s + 7.56·26-s + 2.17·27-s + 5.29·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.50·3-s + 2.25·4-s − 1.62·5-s − 2.72·6-s + 0.443·7-s − 2.26·8-s + 1.27·9-s + 2.93·10-s − 1.29·11-s + 3.40·12-s − 0.822·13-s − 0.800·14-s − 2.45·15-s + 1.82·16-s + 0.247·17-s − 2.30·18-s + 1.87·19-s − 3.67·20-s + 0.669·21-s + 2.33·22-s − 0.238·23-s − 3.41·24-s + 1.65·25-s + 1.48·26-s + 0.418·27-s + 1.00·28-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 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 - 8.16T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.341018063384038524636786536647, −7.84182831168223059162539077807, −7.54351500453160552485163478231, −7.04354159830949987453164187800, −5.34988764188822374366751037773, −4.20805700545660563717792154718, −3.01121533450713389113091574277, −2.70369496495412488697472475801, −1.34864337149636640375650510620, 0,
1.34864337149636640375650510620, 2.70369496495412488697472475801, 3.01121533450713389113091574277, 4.20805700545660563717792154718, 5.34988764188822374366751037773, 7.04354159830949987453164187800, 7.54351500453160552485163478231, 7.84182831168223059162539077807, 8.341018063384038524636786536647
