| L(s) = 1 | + 5.07·2-s − 9.17·3-s + 17.7·4-s + 5·5-s − 46.5·6-s + 27.6·7-s + 49.7·8-s + 57.1·9-s + 25.3·10-s + 11·11-s − 163.·12-s + 61.8·13-s + 140.·14-s − 45.8·15-s + 110.·16-s − 32.8·17-s + 290.·18-s − 19·19-s + 88.9·20-s − 254.·21-s + 55.8·22-s + 45.2·23-s − 455.·24-s + 25·25-s + 314.·26-s − 276.·27-s + 492.·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.76·3-s + 2.22·4-s + 0.447·5-s − 3.16·6-s + 1.49·7-s + 2.19·8-s + 2.11·9-s + 0.802·10-s + 0.301·11-s − 3.92·12-s + 1.32·13-s + 2.68·14-s − 0.789·15-s + 1.72·16-s − 0.468·17-s + 3.80·18-s − 0.229·19-s + 0.994·20-s − 2.63·21-s + 0.541·22-s + 0.410·23-s − 3.87·24-s + 0.200·25-s + 2.37·26-s − 1.97·27-s + 3.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.421336693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.421336693\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 5.07T + 8T^{2} \) |
| 3 | \( 1 + 9.17T + 27T^{2} \) |
| 7 | \( 1 - 27.6T + 343T^{2} \) |
| 13 | \( 1 - 61.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 45.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 134.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 404.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 171.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 866.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 120.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 210.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 91.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 570.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−10.09254977760467424673409941250, −8.538686277021846920919603750273, −7.27852849912949318459301886825, −6.49580879228766135375907765396, −5.84445360733560230352896823283, −5.22827352867269677848308788462, −4.56622334757874393830662182961, −3.79759815722370615811591062379, −2.02943214757651421501841347618, −1.14495932477338168849211295015,
1.14495932477338168849211295015, 2.02943214757651421501841347618, 3.79759815722370615811591062379, 4.56622334757874393830662182961, 5.22827352867269677848308788462, 5.84445360733560230352896823283, 6.49580879228766135375907765396, 7.27852849912949318459301886825, 8.538686277021846920919603750273, 10.09254977760467424673409941250
