| L(s) = 1 | − 0.441·2-s − 3·3-s − 7.80·4-s + 14.6·5-s + 1.32·6-s − 18.4·7-s + 6.97·8-s + 9·9-s − 6.47·10-s + 11·11-s + 23.4·12-s − 46.7·13-s + 8.12·14-s − 44.0·15-s + 59.3·16-s + 52.3·17-s − 3.97·18-s − 105.·19-s − 114.·20-s + 55.2·21-s − 4.85·22-s + 189.·23-s − 20.9·24-s + 90.1·25-s + 20.6·26-s − 27·27-s + 143.·28-s + ⋯ |
| L(s) = 1 | − 0.155·2-s − 0.577·3-s − 0.975·4-s + 1.31·5-s + 0.0900·6-s − 0.995·7-s + 0.308·8-s + 0.333·9-s − 0.204·10-s + 0.301·11-s + 0.563·12-s − 0.997·13-s + 0.155·14-s − 0.757·15-s + 0.927·16-s + 0.746·17-s − 0.0519·18-s − 1.27·19-s − 1.28·20-s + 0.574·21-s − 0.0470·22-s + 1.71·23-s − 0.177·24-s + 0.721·25-s + 0.155·26-s − 0.192·27-s + 0.970·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.441T + 8T^{2} \) |
| 5 | \( 1 - 14.6T + 125T^{2} \) |
| 7 | \( 1 + 18.4T + 343T^{2} \) |
| 13 | \( 1 + 46.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 62.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 31.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 367.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 270.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 603.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 156.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 75.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 778.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−8.749505700353388862356178192507, −7.50015806156921417471783856064, −6.71729975574557874534575464096, −5.89202191582965467716105369897, −5.32452925165705159308602971530, −4.48712346563619005318610124429, −3.41345402614088018610142968077, −2.27014648965051611022337507189, −1.05661625890347102151474677233, 0,
1.05661625890347102151474677233, 2.27014648965051611022337507189, 3.41345402614088018610142968077, 4.48712346563619005318610124429, 5.32452925165705159308602971530, 5.89202191582965467716105369897, 6.71729975574557874534575464096, 7.50015806156921417471783856064, 8.749505700353388862356178192507
