| L(s) = 1 | − 4.64·2-s − 3·3-s + 13.5·4-s − 15.1·5-s + 13.9·6-s + 14.1·7-s − 25.9·8-s + 9·9-s + 70.4·10-s + 11·11-s − 40.7·12-s − 52.0·13-s − 65.5·14-s + 45.4·15-s + 11.9·16-s − 7.88·17-s − 41.8·18-s − 50.5·19-s − 205.·20-s − 42.3·21-s − 51.1·22-s − 220.·23-s + 77.8·24-s + 104.·25-s + 242.·26-s − 27·27-s + 191.·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.577·3-s + 1.69·4-s − 1.35·5-s + 0.948·6-s + 0.761·7-s − 1.14·8-s + 0.333·9-s + 2.22·10-s + 0.301·11-s − 0.980·12-s − 1.11·13-s − 1.25·14-s + 0.782·15-s + 0.186·16-s − 0.112·17-s − 0.547·18-s − 0.610·19-s − 2.30·20-s − 0.439·21-s − 0.495·22-s − 1.99·23-s + 0.662·24-s + 0.838·25-s + 1.82·26-s − 0.192·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09191344296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09191344296\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.64T + 8T^{2} \) |
| 5 | \( 1 + 15.1T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 13 | \( 1 + 52.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.88T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 220.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 64.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 80.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 21.6T + 2.05e5T^{2} \) |
| 67 | \( 1 - 50.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 569.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 9.87T + 3.89e5T^{2} \) |
| 79 | \( 1 + 365.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 595.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.654138939412084272057119758388, −7.948069336198765727620490611734, −7.59718005953318960895060790702, −6.86177379166793996203034329999, −5.85782732117679724546841981353, −4.60078074637999691606997088202, −3.99471509587877877519073112666, −2.41989914531177170080723921586, −1.45075811398081138048113953362, −0.19136714501387914143693471741,
0.19136714501387914143693471741, 1.45075811398081138048113953362, 2.41989914531177170080723921586, 3.99471509587877877519073112666, 4.60078074637999691606997088202, 5.85782732117679724546841981353, 6.86177379166793996203034329999, 7.59718005953318960895060790702, 7.948069336198765727620490611734, 8.654138939412084272057119758388
