| L(s) = 1 | + 3.11·3-s − 4.34·5-s − 1.70·7-s + 6.69·9-s − 2.71·11-s + 5.09·13-s − 13.5·15-s − 2.83·17-s + 6.52·19-s − 5.30·21-s − 5.21·23-s + 13.8·25-s + 11.4·27-s − 2.54·29-s − 4.54·31-s − 8.46·33-s + 7.39·35-s − 1.18·37-s + 15.8·39-s + 7.99·41-s − 3.86·43-s − 29.0·45-s + 2.23·47-s − 4.09·49-s − 8.81·51-s − 3.66·53-s + 11.8·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 1.94·5-s − 0.643·7-s + 2.23·9-s − 0.819·11-s + 1.41·13-s − 3.48·15-s − 0.686·17-s + 1.49·19-s − 1.15·21-s − 1.08·23-s + 2.76·25-s + 2.21·27-s − 0.472·29-s − 0.815·31-s − 1.47·33-s + 1.24·35-s − 0.194·37-s + 2.54·39-s + 1.24·41-s − 0.589·43-s − 4.33·45-s + 0.326·47-s − 0.585·49-s − 1.23·51-s − 0.503·53-s + 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.410869513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410869513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 + 4.34T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 - 4.73T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 8.39T + 89T^{2} \) |
| 97 | \( 1 - 5.39T + 97T^{2} \) |
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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
−7.964909540005309569302447022749, −7.35814013307900872386839659802, −6.92315600569871208388512536663, −5.77655752713267089593742477043, −4.60655950947638491661052805420, −3.94751858888622234939475126992, −3.40874641379005957822115143342, −3.07725171261342881085898369755, −1.98421273626911016312098814528, −0.69145953807147706187472793339,
0.69145953807147706187472793339, 1.98421273626911016312098814528, 3.07725171261342881085898369755, 3.40874641379005957822115143342, 3.94751858888622234939475126992, 4.60655950947638491661052805420, 5.77655752713267089593742477043, 6.92315600569871208388512536663, 7.35814013307900872386839659802, 7.964909540005309569302447022749
