| L(s) = 1 | + 2-s + 2.66·3-s + 4-s + 5-s + 2.66·6-s − 1.80·7-s + 8-s + 4.09·9-s + 10-s + 11-s + 2.66·12-s − 0.569·13-s − 1.80·14-s + 2.66·15-s + 16-s + 5.29·17-s + 4.09·18-s − 0.637·19-s + 20-s − 4.81·21-s + 22-s + 8.02·23-s + 2.66·24-s + 25-s − 0.569·26-s + 2.90·27-s − 1.80·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s + 0.447·5-s + 1.08·6-s − 0.683·7-s + 0.353·8-s + 1.36·9-s + 0.316·10-s + 0.301·11-s + 0.768·12-s − 0.157·13-s − 0.483·14-s + 0.687·15-s + 0.250·16-s + 1.28·17-s + 0.964·18-s − 0.146·19-s + 0.223·20-s − 1.05·21-s + 0.213·22-s + 1.67·23-s + 0.543·24-s + 0.200·25-s − 0.111·26-s + 0.559·27-s − 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.010167260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.010167260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 13 | \( 1 + 0.569T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 0.637T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 6.56T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 - 9.24T + 97T^{2} \) |
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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
−8.296066830495029895015049374683, −7.50995856850521045371232290970, −6.95657402799556624415273240626, −6.11270435239886321438854874983, −5.31249307346525022565293533943, −4.39986841743861188865107535167, −3.42823625374569818951819809181, −3.11117122705125121297046458926, −2.26144993649695532002701087747, −1.25112824214554419542731816001,
1.25112824214554419542731816001, 2.26144993649695532002701087747, 3.11117122705125121297046458926, 3.42823625374569818951819809181, 4.39986841743861188865107535167, 5.31249307346525022565293533943, 6.11270435239886321438854874983, 6.95657402799556624415273240626, 7.50995856850521045371232290970, 8.296066830495029895015049374683
