| L(s) = 1 | + 1.11·3-s + 1.47·5-s + 7-s − 1.75·9-s − 2·11-s − 0.715·13-s + 1.64·15-s − 17-s − 1.28·19-s + 1.11·21-s − 2.22·23-s − 2.83·25-s − 5.30·27-s + 9.17·29-s + 6.75·31-s − 2.22·33-s + 1.47·35-s − 8.45·37-s − 0.798·39-s − 6.77·41-s − 4.39·43-s − 2.58·45-s − 2.22·47-s + 49-s − 1.11·51-s − 6.64·53-s − 2.94·55-s + ⋯ |
| L(s) = 1 | + 0.643·3-s + 0.658·5-s + 0.377·7-s − 0.585·9-s − 0.603·11-s − 0.198·13-s + 0.423·15-s − 0.242·17-s − 0.294·19-s + 0.243·21-s − 0.464·23-s − 0.566·25-s − 1.02·27-s + 1.70·29-s + 1.21·31-s − 0.388·33-s + 0.248·35-s − 1.39·37-s − 0.127·39-s − 1.05·41-s − 0.670·43-s − 0.385·45-s − 0.325·47-s + 0.142·49-s − 0.156·51-s − 0.913·53-s − 0.397·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.715T + 13T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 + 5.28T + 59T^{2} \) |
| 61 | \( 1 + 0.885T + 61T^{2} \) |
| 67 | \( 1 + 4.90T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
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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
−7.82741737601935605806375452680, −6.69683219646470153877371005319, −6.26143603383762404615292062713, −5.29403284134411650899246973738, −4.85700656403506434248818030698, −3.82610190251735905029581365947, −2.93320902731052017577362319596, −2.34493407984308567028685218068, −1.52104452918059723754430541003, 0,
1.52104452918059723754430541003, 2.34493407984308567028685218068, 2.93320902731052017577362319596, 3.82610190251735905029581365947, 4.85700656403506434248818030698, 5.29403284134411650899246973738, 6.26143603383762404615292062713, 6.69683219646470153877371005319, 7.82741737601935605806375452680
