| L(s) = 1 | − 3.06·2-s + 5.71·3-s + 1.37·4-s + 21.1·5-s − 17.5·6-s + 7·7-s + 20.2·8-s + 5.70·9-s − 64.8·10-s + 56.8·11-s + 7.85·12-s − 62.2·13-s − 21.4·14-s + 121.·15-s − 73.0·16-s − 8.75·17-s − 17.4·18-s − 50.6·19-s + 29.1·20-s + 40.0·21-s − 174.·22-s + 23·23-s + 116.·24-s + 324.·25-s + 190.·26-s − 121.·27-s + 9.61·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.10·3-s + 0.171·4-s + 1.89·5-s − 1.19·6-s + 0.377·7-s + 0.896·8-s + 0.211·9-s − 2.05·10-s + 1.55·11-s + 0.188·12-s − 1.32·13-s − 0.409·14-s + 2.08·15-s − 1.14·16-s − 0.124·17-s − 0.228·18-s − 0.611·19-s + 0.325·20-s + 0.415·21-s − 1.68·22-s + 0.208·23-s + 0.986·24-s + 2.59·25-s + 1.43·26-s − 0.868·27-s + 0.0648·28-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
| \(L(2)\) |
\(\approx\) |
\(1.848970353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848970353\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.06T + 8T^{2} \) |
| 3 | \( 1 - 5.71T + 27T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 11 | \( 1 - 56.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.75T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 86.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 86.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 133.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 271.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 777.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 108.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 144.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 155.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 183.T + 9.12e5T^{2} \) |
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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
−12.60916450347538656488655639359, −10.96740004867979464016977829011, −9.792969386610284052142476964920, −9.335056515428196336242186363686, −8.738477378125446450099322102425, −7.46692628352798508138434001924, −6.19755596411333204387998524536, −4.59712605560650156815092956517, −2.49092841387403099856788043743, −1.48196536052837277642657363080,
1.48196536052837277642657363080, 2.49092841387403099856788043743, 4.59712605560650156815092956517, 6.19755596411333204387998524536, 7.46692628352798508138434001924, 8.738477378125446450099322102425, 9.335056515428196336242186363686, 9.792969386610284052142476964920, 10.96740004867979464016977829011, 12.60916450347538656488655639359
