| L(s) = 1 | − 2.35·2-s + 2.82·3-s + 3.53·4-s + 2.18·5-s − 6.64·6-s + 0.649·7-s − 3.60·8-s + 4.99·9-s − 5.14·10-s − 5.34·11-s + 9.98·12-s − 0.189·13-s − 1.52·14-s + 6.18·15-s + 1.40·16-s − 3.29·17-s − 11.7·18-s + 1.35·19-s + 7.72·20-s + 1.83·21-s + 12.5·22-s − 5.91·23-s − 10.1·24-s − 0.219·25-s + 0.446·26-s + 5.62·27-s + 2.29·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.63·3-s + 1.76·4-s + 0.977·5-s − 2.71·6-s + 0.245·7-s − 1.27·8-s + 1.66·9-s − 1.62·10-s − 1.61·11-s + 2.88·12-s − 0.0526·13-s − 0.408·14-s + 1.59·15-s + 0.351·16-s − 0.800·17-s − 2.76·18-s + 0.311·19-s + 1.72·20-s + 0.400·21-s + 2.67·22-s − 1.23·23-s − 2.07·24-s − 0.0438·25-s + 0.0876·26-s + 1.08·27-s + 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 0.649T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 + 0.189T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 0.767T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 0.473T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 + 3.99T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66664077398063002476364664778, −7.41338683761411891143649043514, −6.40967606912533778286906022384, −5.58566609769621904814401196897, −4.60652119648570535066769909978, −3.50316228317238896018678986567, −2.47667512045647059863620795326, −2.22847264403531037037911945050, −1.51809859865364462971779247410, 0,
1.51809859865364462971779247410, 2.22847264403531037037911945050, 2.47667512045647059863620795326, 3.50316228317238896018678986567, 4.60652119648570535066769909978, 5.58566609769621904814401196897, 6.40967606912533778286906022384, 7.41338683761411891143649043514, 7.66664077398063002476364664778
