| L(s) = 1 | − 0.811·2-s − 3-s − 1.34·4-s + 0.350·5-s + 0.811·6-s + 3.06·7-s + 2.71·8-s + 9-s − 0.284·10-s − 11-s + 1.34·12-s − 5.45·13-s − 2.48·14-s − 0.350·15-s + 0.483·16-s − 2.06·17-s − 0.811·18-s + 1.40·19-s − 0.470·20-s − 3.06·21-s + 0.811·22-s + 0.147·23-s − 2.71·24-s − 4.87·25-s + 4.42·26-s − 27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | − 0.573·2-s − 0.577·3-s − 0.670·4-s + 0.156·5-s + 0.331·6-s + 1.15·7-s + 0.958·8-s + 0.333·9-s − 0.0899·10-s − 0.301·11-s + 0.387·12-s − 1.51·13-s − 0.664·14-s − 0.0905·15-s + 0.120·16-s − 0.500·17-s − 0.191·18-s + 0.321·19-s − 0.105·20-s − 0.668·21-s + 0.172·22-s + 0.0308·23-s − 0.553·24-s − 0.975·25-s + 0.868·26-s − 0.192·27-s − 0.777·28-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 | \( 1 + 0.811T + 2T^{2} \) |
| 5 | \( 1 - 0.350T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 - 1.40T + 19T^{2} \) |
| 23 | \( 1 - 0.147T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + 0.144T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 + 4.06T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−8.798217084764634464228837484201, −7.84173756986180013042672180163, −7.58506864422855301502453056694, −6.43465555438310841637561763585, −5.25831265924180523664109464576, −4.86994985130809205052657940260, −4.10781611165146139666994217436, −2.47447574842839354062491128801, −1.36344239379130790527396557956, 0,
1.36344239379130790527396557956, 2.47447574842839354062491128801, 4.10781611165146139666994217436, 4.86994985130809205052657940260, 5.25831265924180523664109464576, 6.43465555438310841637561763585, 7.58506864422855301502453056694, 7.84173756986180013042672180163, 8.798217084764634464228837484201
