| L(s) = 1 | − 1.87·2-s − 2.61·3-s + 1.53·4-s − 1.23·5-s + 4.90·6-s − 3.71·7-s + 0.878·8-s + 3.82·9-s + 2.32·10-s − 11-s − 4.00·12-s − 4.16·13-s + 6.98·14-s + 3.22·15-s − 4.71·16-s + 17-s − 7.18·18-s − 5.54·19-s − 1.89·20-s + 9.70·21-s + 1.87·22-s − 0.942·23-s − 2.29·24-s − 3.47·25-s + 7.83·26-s − 2.14·27-s − 5.69·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 1.50·3-s + 0.766·4-s − 0.552·5-s + 2.00·6-s − 1.40·7-s + 0.310·8-s + 1.27·9-s + 0.734·10-s − 0.301·11-s − 1.15·12-s − 1.15·13-s + 1.86·14-s + 0.833·15-s − 1.17·16-s + 0.242·17-s − 1.69·18-s − 1.27·19-s − 0.423·20-s + 2.11·21-s + 0.400·22-s − 0.196·23-s − 0.468·24-s − 0.694·25-s + 1.53·26-s − 0.412·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 0.942T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 9.01T + 37T^{2} \) |
| 41 | \( 1 - 1.96T + 41T^{2} \) |
| 47 | \( 1 + 4.72T + 47T^{2} \) |
| 53 | \( 1 + 7.71T + 53T^{2} \) |
| 59 | \( 1 - 9.28T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 - 0.378T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−7.59862284381547854866630986923, −6.79792674847671930728001672354, −6.31792258715173947724129036542, −5.63137671548414809348188078619, −4.67572784867519056245080839088, −4.10672403959210829912240242453, −2.92294531089646135990141379902, −1.87303305943635310292702496507, −0.52493650453488624892835294245, 0,
0.52493650453488624892835294245, 1.87303305943635310292702496507, 2.92294531089646135990141379902, 4.10672403959210829912240242453, 4.67572784867519056245080839088, 5.63137671548414809348188078619, 6.31792258715173947724129036542, 6.79792674847671930728001672354, 7.59862284381547854866630986923
