| L(s) = 1 | − 2-s − 1.70·3-s + 4-s + 4.07·5-s + 1.70·6-s + 1.03·7-s − 8-s − 0.101·9-s − 4.07·10-s − 1.31·11-s − 1.70·12-s + 2.37·13-s − 1.03·14-s − 6.93·15-s + 16-s − 5.13·17-s + 0.101·18-s + 4.46·19-s + 4.07·20-s − 1.76·21-s + 1.31·22-s + 0.760·23-s + 1.70·24-s + 11.6·25-s − 2.37·26-s + 5.28·27-s + 1.03·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.982·3-s + 0.5·4-s + 1.82·5-s + 0.694·6-s + 0.392·7-s − 0.353·8-s − 0.0339·9-s − 1.28·10-s − 0.395·11-s − 0.491·12-s + 0.657·13-s − 0.277·14-s − 1.79·15-s + 0.250·16-s − 1.24·17-s + 0.0240·18-s + 1.02·19-s + 0.911·20-s − 0.385·21-s + 0.279·22-s + 0.158·23-s + 0.347·24-s + 2.32·25-s − 0.465·26-s + 1.01·27-s + 0.196·28-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
| \(L(1)\) |
\(\approx\) |
\(1.504633658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504633658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 0.760T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 + 0.698T + 31T^{2} \) |
| 37 | \( 1 + 7.91T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 + 0.495T + 53T^{2} \) |
| 59 | \( 1 + 1.42T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 + 8.35T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−7.85730200626778989645293321200, −6.92829112232449511488557748560, −6.32997287349651728614712114977, −5.94214608676128197870490570133, −5.18001390032135115726487258690, −4.74676322625385076986878850030, −3.22019829811895963849480329579, −2.37515108222371391586173554794, −1.61494018511701429620088647799, −0.73790841384244003130361911806,
0.73790841384244003130361911806, 1.61494018511701429620088647799, 2.37515108222371391586173554794, 3.22019829811895963849480329579, 4.74676322625385076986878850030, 5.18001390032135115726487258690, 5.94214608676128197870490570133, 6.32997287349651728614712114977, 6.92829112232449511488557748560, 7.85730200626778989645293321200
