| L(s) = 1 | + 5.07·2-s − 3·3-s + 17.7·4-s + 3.68·5-s − 15.2·6-s − 36.3·7-s + 49.7·8-s + 9·9-s + 18.7·10-s − 11·11-s − 53.3·12-s + 80.4·13-s − 184.·14-s − 11.0·15-s + 110.·16-s + 77.1·17-s + 45.7·18-s − 38.6·19-s + 65.6·20-s + 108.·21-s − 55.8·22-s − 184.·23-s − 149.·24-s − 111.·25-s + 408.·26-s − 27·27-s − 646.·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.22·4-s + 0.330·5-s − 1.03·6-s − 1.96·7-s + 2.19·8-s + 0.333·9-s + 0.592·10-s − 0.301·11-s − 1.28·12-s + 1.71·13-s − 3.52·14-s − 0.190·15-s + 1.72·16-s + 1.10·17-s + 0.598·18-s − 0.466·19-s + 0.734·20-s + 1.13·21-s − 0.541·22-s − 1.67·23-s − 1.26·24-s − 0.891·25-s + 3.08·26-s − 0.192·27-s − 4.36·28-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 | \( 1 - 5.07T + 8T^{2} \) |
| 5 | \( 1 - 3.68T + 125T^{2} \) |
| 7 | \( 1 + 36.3T + 343T^{2} \) |
| 13 | \( 1 - 80.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 52.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 266.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 574.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 600.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 75.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 184.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 349.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 96.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 757.T + 9.12e5T^{2} \) |
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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
−8.177240216414403142960948564328, −7.07402865791789780862886585347, −6.34453535138059716755966604218, −5.89624682423514252520992404313, −5.51156401143333893206382288182, −4.01539536376901369333627235253, −3.70373618973601286348975552814, −2.82336044476494747225258053030, −1.61745901375149920046705535944, 0,
1.61745901375149920046705535944, 2.82336044476494747225258053030, 3.70373618973601286348975552814, 4.01539536376901369333627235253, 5.51156401143333893206382288182, 5.89624682423514252520992404313, 6.34453535138059716755966604218, 7.07402865791789780862886585347, 8.177240216414403142960948564328
