| L(s) = 1 | + 3.76·2-s + 2.58·3-s + 6.19·4-s + 21.4·5-s + 9.73·6-s + 7·7-s − 6.79·8-s − 20.3·9-s + 80.7·10-s − 34.1·11-s + 16.0·12-s + 66.8·13-s + 26.3·14-s + 55.4·15-s − 75.1·16-s − 107.·17-s − 76.5·18-s + 33.1·19-s + 132.·20-s + 18.0·21-s − 128.·22-s + 23·23-s − 17.5·24-s + 334.·25-s + 251.·26-s − 122.·27-s + 43.3·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.497·3-s + 0.774·4-s + 1.91·5-s + 0.662·6-s + 0.377·7-s − 0.300·8-s − 0.752·9-s + 2.55·10-s − 0.937·11-s + 0.385·12-s + 1.42·13-s + 0.503·14-s + 0.953·15-s − 1.17·16-s − 1.52·17-s − 1.00·18-s + 0.400·19-s + 1.48·20-s + 0.188·21-s − 1.24·22-s + 0.208·23-s − 0.149·24-s + 2.67·25-s + 1.89·26-s − 0.871·27-s + 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.523246107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.523246107\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3.76T + 8T^{2} \) |
| 3 | \( 1 - 2.58T + 27T^{2} \) |
| 5 | \( 1 - 21.4T + 125T^{2} \) |
| 11 | \( 1 + 34.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 24.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 76.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 221.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 761.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 756.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 847.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 453.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 3.62T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 94.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 781.T + 9.12e5T^{2} \) |
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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
−13.17834925249602604098028881582, −11.53187921855638230298925546323, −10.57607554806598007495587799391, −9.251908255656217762482303437646, −8.487120192738403543369389689803, −6.48266315227105248797507769313, −5.76211204149187667010708561872, −4.83435423740997323000867591672, −3.09529453545664900382723966169, −2.06590485420873796641564444998,
2.06590485420873796641564444998, 3.09529453545664900382723966169, 4.83435423740997323000867591672, 5.76211204149187667010708561872, 6.48266315227105248797507769313, 8.487120192738403543369389689803, 9.251908255656217762482303437646, 10.57607554806598007495587799391, 11.53187921855638230298925546323, 13.17834925249602604098028881582
