| L(s) = 1 | + 4.07·2-s + 5.92·3-s + 8.58·4-s + 17.4·5-s + 24.1·6-s − 2.87·7-s + 2.38·8-s + 8.16·9-s + 71.0·10-s − 7.41·11-s + 50.9·12-s + 48.1·13-s − 11.7·14-s + 103.·15-s − 58.9·16-s + 45.3·17-s + 33.2·18-s + 37.7·19-s + 149.·20-s − 17.0·21-s − 30.1·22-s − 23·23-s + 14.1·24-s + 179.·25-s + 195.·26-s − 111.·27-s − 24.6·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.14·3-s + 1.07·4-s + 1.56·5-s + 1.64·6-s − 0.155·7-s + 0.105·8-s + 0.302·9-s + 2.24·10-s − 0.203·11-s + 1.22·12-s + 1.02·13-s − 0.223·14-s + 1.78·15-s − 0.921·16-s + 0.646·17-s + 0.435·18-s + 0.455·19-s + 1.67·20-s − 0.177·21-s − 0.292·22-s − 0.208·23-s + 0.120·24-s + 1.43·25-s + 1.47·26-s − 0.796·27-s − 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.229232114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.229232114\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.07T + 8T^{2} \) |
| 3 | \( 1 - 5.92T + 27T^{2} \) |
| 5 | \( 1 - 17.4T + 125T^{2} \) |
| 7 | \( 1 + 2.87T + 343T^{2} \) |
| 11 | \( 1 + 7.41T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.7T + 6.85e3T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 427.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 244.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 375.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 404.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 546.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 221.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 466.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 803.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−10.02633238850206647800754201129, −9.226450934690063533848691986099, −8.544629258804253093385035123733, −7.28519703903300284164421810112, −6.00222698761416519471173956106, −5.76958248772508619744743247192, −4.50675287816442252222303411618, −3.31664367973212197608368988651, −2.70813552595949027936843306532, −1.60379544141556050401722222130,
1.60379544141556050401722222130, 2.70813552595949027936843306532, 3.31664367973212197608368988651, 4.50675287816442252222303411618, 5.76958248772508619744743247192, 6.00222698761416519471173956106, 7.28519703903300284164421810112, 8.544629258804253093385035123733, 9.226450934690063533848691986099, 10.02633238850206647800754201129
