| L(s) = 1 | + 2.39·2-s − 3-s + 3.71·4-s − 2.34·5-s − 2.39·6-s − 0.680·7-s + 4.10·8-s + 9-s − 5.59·10-s + 11-s − 3.71·12-s − 4.08·13-s − 1.62·14-s + 2.34·15-s + 2.38·16-s − 4.92·17-s + 2.39·18-s − 2.37·19-s − 8.70·20-s + 0.680·21-s + 2.39·22-s + 8.55·23-s − 4.10·24-s + 0.485·25-s − 9.76·26-s − 27-s − 2.53·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.85·4-s − 1.04·5-s − 0.976·6-s − 0.257·7-s + 1.45·8-s + 0.333·9-s − 1.77·10-s + 0.301·11-s − 1.07·12-s − 1.13·13-s − 0.435·14-s + 0.604·15-s + 0.595·16-s − 1.19·17-s + 0.563·18-s − 0.545·19-s − 1.94·20-s + 0.148·21-s + 0.509·22-s + 1.78·23-s − 0.837·24-s + 0.0970·25-s − 1.91·26-s − 0.192·27-s − 0.478·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 0.680T + 7T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 - 8.55T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 - 4.84T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.142T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 8.41T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−8.668798721970123235582677982382, −7.55256477884753595250596419897, −6.79875163488084367517119510195, −6.45542139548641899787464346730, −5.08129564762366532850755708837, −4.87311046890843523080462118824, −3.90571786092621634143740518771, −3.20628576425907611425832402693, −2.04079537719894268774447475633, 0,
2.04079537719894268774447475633, 3.20628576425907611425832402693, 3.90571786092621634143740518771, 4.87311046890843523080462118824, 5.08129564762366532850755708837, 6.45542139548641899787464346730, 6.79875163488084367517119510195, 7.55256477884753595250596419897, 8.668798721970123235582677982382
