| L(s) = 1 | + 2-s + 0.675·3-s + 4-s + 1.23·5-s + 0.675·6-s + 0.920·7-s + 8-s − 2.54·9-s + 1.23·10-s + 0.877·11-s + 0.675·12-s − 3.47·13-s + 0.920·14-s + 0.836·15-s + 16-s + 0.808·17-s − 2.54·18-s − 6.30·19-s + 1.23·20-s + 0.621·21-s + 0.877·22-s − 4.48·23-s + 0.675·24-s − 3.46·25-s − 3.47·26-s − 3.74·27-s + 0.920·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.389·3-s + 0.5·4-s + 0.554·5-s + 0.275·6-s + 0.347·7-s + 0.353·8-s − 0.847·9-s + 0.391·10-s + 0.264·11-s + 0.194·12-s − 0.962·13-s + 0.245·14-s + 0.216·15-s + 0.250·16-s + 0.196·17-s − 0.599·18-s − 1.44·19-s + 0.277·20-s + 0.135·21-s + 0.187·22-s − 0.935·23-s + 0.137·24-s − 0.692·25-s − 0.680·26-s − 0.720·27-s + 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 0.675T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 0.920T + 7T^{2} \) |
| 11 | \( 1 - 0.877T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 0.808T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 - 0.0488T + 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 2.63T + 73T^{2} \) |
| 79 | \( 1 - 4.91T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.71525816405036011510969922579, −6.93816767802941839470532324308, −6.12988406103169083378969427927, −5.58927321112336132815174850085, −4.87894469268893724363558735902, −4.00035175489210022972987811126, −3.29971923257641294547469218621, −2.17734325166619085009213257766, −1.93005111777150764216160705862, 0,
1.93005111777150764216160705862, 2.17734325166619085009213257766, 3.29971923257641294547469218621, 4.00035175489210022972987811126, 4.87894469268893724363558735902, 5.58927321112336132815174850085, 6.12988406103169083378969427927, 6.93816767802941839470532324308, 7.71525816405036011510969922579
