| L(s) = 1 | − 3-s + 5-s − 2.56·7-s + 9-s − 1.43·11-s + 13-s − 15-s + 5.68·17-s − 5.12·19-s + 2.56·21-s − 1.43·23-s + 25-s − 27-s − 2·29-s + 1.12·31-s + 1.43·33-s − 2.56·35-s − 10.8·37-s − 39-s − 9.68·41-s + 6.24·43-s + 45-s − 1.12·47-s − 0.438·49-s − 5.68·51-s − 0.561·53-s − 1.43·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.968·7-s + 0.333·9-s − 0.433·11-s + 0.277·13-s − 0.258·15-s + 1.37·17-s − 1.17·19-s + 0.558·21-s − 0.299·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s + 0.201·31-s + 0.250·33-s − 0.432·35-s − 1.77·37-s − 0.160·39-s − 1.51·41-s + 0.952·43-s + 0.149·45-s − 0.163·47-s − 0.0626·49-s − 0.796·51-s − 0.0771·53-s − 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−9.129052017223740455350397482064, −8.284619757135976812825184874847, −7.27963111380182083595123314937, −6.47555709008817504279660443924, −5.82421214765066569221039474000, −5.06930188695849626713046925189, −3.87483087660444351327862411011, −2.95554076281844851911141349426, −1.61645514247363549630957226429, 0,
1.61645514247363549630957226429, 2.95554076281844851911141349426, 3.87483087660444351327862411011, 5.06930188695849626713046925189, 5.82421214765066569221039474000, 6.47555709008817504279660443924, 7.27963111380182083595123314937, 8.284619757135976812825184874847, 9.129052017223740455350397482064
