| L(s) = 1 | + 2.19·3-s + 0.455·5-s − 2.54·7-s + 1.81·9-s + 0.262·11-s − 5.81·13-s + 0.999·15-s + 2.25·17-s + 7.64·19-s − 5.59·21-s − 2.29·23-s − 4.79·25-s − 2.60·27-s − 7.68·29-s − 6.67·31-s + 0.575·33-s − 1.16·35-s − 4.67·37-s − 12.7·39-s − 6.97·41-s + 9.67·43-s + 0.825·45-s − 0.331·47-s − 0.501·49-s + 4.94·51-s + 4.11·53-s + 0.119·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.203·5-s − 0.963·7-s + 0.603·9-s + 0.0791·11-s − 1.61·13-s + 0.258·15-s + 0.547·17-s + 1.75·19-s − 1.22·21-s − 0.478·23-s − 0.958·25-s − 0.501·27-s − 1.42·29-s − 1.19·31-s + 0.100·33-s − 0.196·35-s − 0.769·37-s − 2.04·39-s − 1.08·41-s + 1.47·43-s + 0.123·45-s − 0.0483·47-s − 0.0716·49-s + 0.692·51-s + 0.565·53-s + 0.0161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 0.455T + 5T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 11 | \( 1 - 0.262T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 6.97T + 41T^{2} \) |
| 43 | \( 1 - 9.67T + 43T^{2} \) |
| 47 | \( 1 + 0.331T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 - 8.03T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 + 7.06T + 89T^{2} \) |
| 97 | \( 1 - 1.60T + 97T^{2} \) |
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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
−7.84437054274685875121986715935, −7.58403769062038651893033149401, −6.87815954484405940164561897113, −5.72230420754594599714992846861, −5.21125902035441335408612697128, −3.87603489695841317312055477176, −3.35786880508558534559026705260, −2.58698286278550430157717957730, −1.75149982575177187027476539056, 0,
1.75149982575177187027476539056, 2.58698286278550430157717957730, 3.35786880508558534559026705260, 3.87603489695841317312055477176, 5.21125902035441335408612697128, 5.72230420754594599714992846861, 6.87815954484405940164561897113, 7.58403769062038651893033149401, 7.84437054274685875121986715935
