| L(s) = 1 | − 1.36·3-s + 2·5-s + 1.42·7-s − 1.14·9-s − 2.19·11-s + 1.74·13-s − 2.72·15-s + 3.26·17-s − 5.89·19-s − 1.94·21-s − 6.12·23-s − 25-s + 5.64·27-s − 0.321·29-s − 1.75·31-s + 2.98·33-s + 2.85·35-s − 2.47·37-s − 2.38·39-s − 5.98·41-s + 3.49·43-s − 2.28·45-s − 2.39·47-s − 4.96·49-s − 4.44·51-s − 1.51·53-s − 4.38·55-s + ⋯ |
| L(s) = 1 | − 0.786·3-s + 0.894·5-s + 0.539·7-s − 0.381·9-s − 0.660·11-s + 0.484·13-s − 0.703·15-s + 0.791·17-s − 1.35·19-s − 0.424·21-s − 1.27·23-s − 0.200·25-s + 1.08·27-s − 0.0596·29-s − 0.314·31-s + 0.519·33-s + 0.482·35-s − 0.406·37-s − 0.381·39-s − 0.935·41-s + 0.533·43-s − 0.341·45-s − 0.349·47-s − 0.709·49-s − 0.622·51-s − 0.208·53-s − 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 + 0.321T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + 2.39T + 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 9.61T + 67T^{2} \) |
| 71 | \( 1 - 6.66T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 + 0.256T + 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−8.335774464229008813782650813316, −7.897993580616175253128806343624, −6.65438720231523630057062166583, −6.03814136431425357284350306734, −5.48657831211420424987016782383, −4.77778934128581720894033377459, −3.67182373723429727332017690344, −2.43940606212947957932076026026, −1.56164088792158359296182374027, 0,
1.56164088792158359296182374027, 2.43940606212947957932076026026, 3.67182373723429727332017690344, 4.77778934128581720894033377459, 5.48657831211420424987016782383, 6.03814136431425357284350306734, 6.65438720231523630057062166583, 7.897993580616175253128806343624, 8.335774464229008813782650813316
