| L(s) = 1 | − 2-s + 4-s − 0.267·5-s + 0.732·7-s − 8-s + 0.267·10-s − 4.73·11-s − 0.732·14-s + 16-s + 2.26·17-s + 1.26·19-s − 0.267·20-s + 4.73·22-s + 6.19·23-s − 4.92·25-s + 0.732·28-s − 2.46·29-s − 5.46·31-s − 32-s − 2.26·34-s − 0.196·35-s + 10.4·37-s − 1.26·38-s + 0.267·40-s − 11.3·41-s + 7.66·43-s − 4.73·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.119·5-s + 0.276·7-s − 0.353·8-s + 0.0847·10-s − 1.42·11-s − 0.195·14-s + 0.250·16-s + 0.550·17-s + 0.290·19-s − 0.0599·20-s + 1.00·22-s + 1.29·23-s − 0.985·25-s + 0.138·28-s − 0.457·29-s − 0.981·31-s − 0.176·32-s − 0.388·34-s − 0.0331·35-s + 1.72·37-s − 0.205·38-s + 0.0423·40-s − 1.77·41-s + 1.16·43-s − 0.713·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.267T + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 9.73T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.295142690441896045838102616374, −7.55448377230103792037925796389, −7.27348662695797111908308120222, −5.99008302539566747478478349696, −5.42271508172174471658740215124, −4.50195461608258171641224450849, −3.29815734924022765347393987105, −2.51942714308908072290373260202, −1.37785165949557012853961381382, 0,
1.37785165949557012853961381382, 2.51942714308908072290373260202, 3.29815734924022765347393987105, 4.50195461608258171641224450849, 5.42271508172174471658740215124, 5.99008302539566747478478349696, 7.27348662695797111908308120222, 7.55448377230103792037925796389, 8.295142690441896045838102616374
