| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 7.50·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 15.0·10-s + 52.6·11-s + 12·12-s − 77.2·13-s + 14·14-s + 22.5·15-s + 16·16-s − 25.2·17-s − 18·18-s + 98.1·19-s + 30.0·20-s − 21·21-s − 105.·22-s + 23·23-s − 24·24-s − 68.6·25-s + 154.·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.671·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.474·10-s + 1.44·11-s + 0.288·12-s − 1.64·13-s + 0.267·14-s + 0.387·15-s + 0.250·16-s − 0.359·17-s − 0.235·18-s + 1.18·19-s + 0.335·20-s − 0.218·21-s − 1.02·22-s + 0.208·23-s − 0.204·24-s − 0.548·25-s + 1.16·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.137905694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137905694\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 7.50T + 125T^{2} \) |
| 11 | \( 1 - 52.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 322.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 32.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 1.66T + 2.05e5T^{2} \) |
| 61 | \( 1 + 431.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 762.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 901.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 249.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 622.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 85.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−9.520762796413776319543970351954, −9.135393149076173731954591625370, −8.017397444819439383876112967348, −7.13618770402582925990197709726, −6.52194427821821100269360599590, −5.40392774208804258343354887413, −4.18300393588274140633298976958, −2.95336039552095773693697775907, −2.06261041444644822860930784194, −0.865011867652028761010075338142,
0.865011867652028761010075338142, 2.06261041444644822860930784194, 2.95336039552095773693697775907, 4.18300393588274140633298976958, 5.40392774208804258343354887413, 6.52194427821821100269360599590, 7.13618770402582925990197709726, 8.017397444819439383876112967348, 9.135393149076173731954591625370, 9.520762796413776319543970351954
