L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 5·11-s − 5·13-s + 16-s + 4·17-s − 7·19-s − 20-s − 5·22-s − 23-s + 25-s + 5·26-s − 2·31-s − 32-s − 4·34-s + 37-s + 7·38-s + 40-s − 5·41-s + 12·43-s + 5·44-s + 46-s + 11·47-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 1.38·13-s + 1/4·16-s + 0.970·17-s − 1.60·19-s − 0.223·20-s − 1.06·22-s − 0.208·23-s + 1/5·25-s + 0.980·26-s − 0.359·31-s − 0.176·32-s − 0.685·34-s + 0.164·37-s + 1.13·38-s + 0.158·40-s − 0.780·41-s + 1.82·43-s + 0.753·44-s + 0.147·46-s + 1.60·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107026500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107026500\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.484879927600426097912628351699, −7.53787161784493367800242626257, −7.14649750147842039092805959768, −6.32540875064171518411237342742, −5.57775317661016459216005230455, −4.42280402773916569966283788278, −3.87080275979738045022621798542, −2.74486986379850522722257339751, −1.83135197960796867606920027662, −0.65591711457808033856547639345,
0.65591711457808033856547639345, 1.83135197960796867606920027662, 2.74486986379850522722257339751, 3.87080275979738045022621798542, 4.42280402773916569966283788278, 5.57775317661016459216005230455, 6.32540875064171518411237342742, 7.14649750147842039092805959768, 7.53787161784493367800242626257, 8.484879927600426097912628351699