| L(s) = 1 | − 1.24·2-s + 2.17·3-s − 0.454·4-s − 1.92·5-s − 2.70·6-s + 3.86·7-s + 3.05·8-s + 1.73·9-s + 2.39·10-s + 6.39·11-s − 0.988·12-s + 5.89·13-s − 4.79·14-s − 4.18·15-s − 2.88·16-s + 0.389·17-s − 2.15·18-s + 4.73·19-s + 0.874·20-s + 8.39·21-s − 7.94·22-s − 2.02·23-s + 6.63·24-s − 1.29·25-s − 7.32·26-s − 2.75·27-s − 1.75·28-s + ⋯ |
| L(s) = 1 | − 0.879·2-s + 1.25·3-s − 0.227·4-s − 0.860·5-s − 1.10·6-s + 1.45·7-s + 1.07·8-s + 0.578·9-s + 0.756·10-s + 1.92·11-s − 0.285·12-s + 1.63·13-s − 1.28·14-s − 1.08·15-s − 0.721·16-s + 0.0944·17-s − 0.508·18-s + 1.08·19-s + 0.195·20-s + 1.83·21-s − 1.69·22-s − 0.422·23-s + 1.35·24-s − 0.259·25-s − 1.43·26-s − 0.530·27-s − 0.331·28-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
| \(L(1)\) |
\(\approx\) |
\(2.076660666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076660666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 0.389T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 0.283T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.784735193249689916492080306581, −8.239094054223942436943256554495, −7.79267229438957936671188477144, −7.03225644645841234553172605082, −5.83662124913737038481896575672, −4.48706611890208405374090760354, −4.02995316566441300660041546002, −3.27857502296822411994530620725, −1.64775585043416899228084803853, −1.16856199396226486140564101009,
1.16856199396226486140564101009, 1.64775585043416899228084803853, 3.27857502296822411994530620725, 4.02995316566441300660041546002, 4.48706611890208405374090760354, 5.83662124913737038481896575672, 7.03225644645841234553172605082, 7.79267229438957936671188477144, 8.239094054223942436943256554495, 8.784735193249689916492080306581
