L(s) = 1 | + 2-s + 1.73·3-s + 4-s − 5-s + 1.73·6-s + 0.732·7-s + 8-s − 10-s − 3.46·11-s + 1.73·12-s + 0.732·14-s − 1.73·15-s + 16-s − 3.73·17-s − 4.46·19-s − 20-s + 1.26·21-s − 3.46·22-s + 1.73·24-s + 25-s − 5.19·27-s + 0.732·28-s − 0.732·29-s − 1.73·30-s + 0.196·31-s + 32-s − 5.99·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.707·6-s + 0.276·7-s + 0.353·8-s − 0.316·10-s − 1.04·11-s + 0.500·12-s + 0.195·14-s − 0.447·15-s + 0.250·16-s − 0.905·17-s − 1.02·19-s − 0.223·20-s + 0.276·21-s − 0.738·22-s + 0.353·24-s + 0.200·25-s − 1.00·27-s + 0.138·28-s − 0.135·29-s − 0.316·30-s + 0.0352·31-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.19T + 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 1.73T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 2.19T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−7.889380013817030033532315537819, −7.17654556680826049700471754824, −6.45787930808684760345840156059, −5.48998804059451586097240085382, −4.81354510472254401576218933202, −4.02089084361273029699273033823, −3.29494968027055432555737442050, −2.50811855396218150657590962044, −1.85762107957747586034295888855, 0,
1.85762107957747586034295888855, 2.50811855396218150657590962044, 3.29494968027055432555737442050, 4.02089084361273029699273033823, 4.81354510472254401576218933202, 5.48998804059451586097240085382, 6.45787930808684760345840156059, 7.17654556680826049700471754824, 7.889380013817030033532315537819