L(s) = 1 | − 2-s − 3.14·3-s + 4-s + 3.08·5-s + 3.14·6-s + 2.61·7-s − 8-s + 6.89·9-s − 3.08·10-s + 3.38·11-s − 3.14·12-s + 4.00·13-s − 2.61·14-s − 9.70·15-s + 16-s + 0.238·17-s − 6.89·18-s + 0.0258·19-s + 3.08·20-s − 8.21·21-s − 3.38·22-s + 2.40·23-s + 3.14·24-s + 4.50·25-s − 4.00·26-s − 12.2·27-s + 2.61·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.81·3-s + 0.5·4-s + 1.37·5-s + 1.28·6-s + 0.986·7-s − 0.353·8-s + 2.29·9-s − 0.975·10-s + 1.02·11-s − 0.908·12-s + 1.11·13-s − 0.697·14-s − 2.50·15-s + 0.250·16-s + 0.0579·17-s − 1.62·18-s + 0.00593·19-s + 0.689·20-s − 1.79·21-s − 0.722·22-s + 0.501·23-s + 0.642·24-s + 0.901·25-s − 0.785·26-s − 2.36·27-s + 0.493·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495523733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495523733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 17 | \( 1 - 0.238T + 17T^{2} \) |
| 19 | \( 1 - 0.0258T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 7.95T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 + 0.667T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 5.12T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 + 0.921T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 + 2.35T + 83T^{2} \) |
| 89 | \( 1 - 5.42T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−7.62596273407582432386942715064, −6.99775253822426218014876419443, −6.26312183240833184791336081768, −5.90379978188633074288857575646, −5.33785064749760288485063223732, −4.57373979636297408149936191584, −3.66108767924854966469751147588, −2.06797816489983142476237034540, −1.45685490024656376973067185590, −0.847162676988962074969343267052,
0.847162676988962074969343267052, 1.45685490024656376973067185590, 2.06797816489983142476237034540, 3.66108767924854966469751147588, 4.57373979636297408149936191584, 5.33785064749760288485063223732, 5.90379978188633074288857575646, 6.26312183240833184791336081768, 6.99775253822426218014876419443, 7.62596273407582432386942715064