| L(s) = 1 | + 0.439·2-s + 3·3-s − 7.80·4-s − 18.8·5-s + 1.31·6-s − 35.5·7-s − 6.95·8-s + 9·9-s − 8.30·10-s + 11·11-s − 23.4·12-s + 49.3·13-s − 15.6·14-s − 56.6·15-s + 59.3·16-s − 110.·17-s + 3.95·18-s + 71.2·19-s + 147.·20-s − 106.·21-s + 4.83·22-s − 66.5·23-s − 20.8·24-s + 231.·25-s + 21.7·26-s + 27·27-s + 277.·28-s + ⋯ |
| L(s) = 1 | + 0.155·2-s + 0.577·3-s − 0.975·4-s − 1.68·5-s + 0.0897·6-s − 1.91·7-s − 0.307·8-s + 0.333·9-s − 0.262·10-s + 0.301·11-s − 0.563·12-s + 1.05·13-s − 0.298·14-s − 0.974·15-s + 0.928·16-s − 1.57·17-s + 0.0518·18-s + 0.859·19-s + 1.64·20-s − 1.10·21-s + 0.0468·22-s − 0.603·23-s − 0.177·24-s + 1.85·25-s + 0.163·26-s + 0.192·27-s + 1.87·28-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 | \( 1 - 0.439T + 8T^{2} \) |
| 5 | \( 1 + 18.8T + 125T^{2} \) |
| 7 | \( 1 + 35.5T + 343T^{2} \) |
| 13 | \( 1 - 49.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 535.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 574.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 305.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 548.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 428.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 335.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.54e3T + 9.12e5T^{2} \) |
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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
−8.370476154923061413556114954212, −7.931606843248005109256792043571, −6.59980111079211645115111823544, −6.42904977331512262929425674702, −4.78375218341626629186824580449, −4.11078518004144118104353750593, −3.44349382064713472833820530220, −2.95254270622752096945318436378, −0.848416944693612690873217972371, 0,
0.848416944693612690873217972371, 2.95254270622752096945318436378, 3.44349382064713472833820530220, 4.11078518004144118104353750593, 4.78375218341626629186824580449, 6.42904977331512262929425674702, 6.59980111079211645115111823544, 7.931606843248005109256792043571, 8.370476154923061413556114954212
