| L(s) = 1 | + 1.13·3-s − 1.24·5-s − 4.57·7-s − 1.70·9-s − 0.731·11-s + 1.94·13-s − 1.41·15-s + 4.32·17-s + 0.162·19-s − 5.20·21-s + 9.50·23-s − 3.44·25-s − 5.34·27-s + 8.98·29-s + 4.60·31-s − 0.830·33-s + 5.70·35-s − 1.75·37-s + 2.20·39-s − 0.0325·41-s − 8.64·43-s + 2.13·45-s − 11.4·47-s + 13.9·49-s + 4.91·51-s + 6.31·53-s + 0.911·55-s + ⋯ |
| L(s) = 1 | + 0.655·3-s − 0.557·5-s − 1.73·7-s − 0.569·9-s − 0.220·11-s + 0.539·13-s − 0.365·15-s + 1.04·17-s + 0.0372·19-s − 1.13·21-s + 1.98·23-s − 0.689·25-s − 1.02·27-s + 1.66·29-s + 0.827·31-s − 0.144·33-s + 0.964·35-s − 0.288·37-s + 0.353·39-s − 0.00508·41-s − 1.31·43-s + 0.317·45-s − 1.67·47-s + 1.99·49-s + 0.688·51-s + 0.867·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 0.731T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 0.162T + 19T^{2} \) |
| 23 | \( 1 - 9.50T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 0.0325T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 + 0.425T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 8.07T + 97T^{2} \) |
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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
−7.47390201540781413941122884066, −6.82408068010247177282329235960, −6.21065500901639677784529280882, −5.48242587954956480860410409474, −4.56932480439754710375324244103, −3.46848400741238863713120856289, −3.22403113044331038624321453502, −2.64350343747352605854038205949, −1.13453812161639873126815274271, 0,
1.13453812161639873126815274271, 2.64350343747352605854038205949, 3.22403113044331038624321453502, 3.46848400741238863713120856289, 4.56932480439754710375324244103, 5.48242587954956480860410409474, 6.21065500901639677784529280882, 6.82408068010247177282329235960, 7.47390201540781413941122884066
