| L(s) = 1 | − 2.61·2-s − 2.61·3-s + 4.85·4-s − 2.61·5-s + 6.85·6-s − 7-s − 7.47·8-s + 3.85·9-s + 6.85·10-s − 1.85·11-s − 12.7·12-s + 2.61·14-s + 6.85·15-s + 9.85·16-s − 1.47·17-s − 10.0·18-s + 1.85·19-s − 12.7·20-s + 2.61·21-s + 4.85·22-s − 4.47·23-s + 19.5·24-s + 1.85·25-s − 2.23·27-s − 4.85·28-s + 7.09·29-s − 17.9·30-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.51·3-s + 2.42·4-s − 1.17·5-s + 2.79·6-s − 0.377·7-s − 2.64·8-s + 1.28·9-s + 2.16·10-s − 0.559·11-s − 3.66·12-s + 0.699·14-s + 1.76·15-s + 2.46·16-s − 0.357·17-s − 2.37·18-s + 0.425·19-s − 2.84·20-s + 0.571·21-s + 1.03·22-s − 0.932·23-s + 3.99·24-s + 0.370·25-s − 0.430·27-s − 0.917·28-s + 1.31·29-s − 3.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
| 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
−9.513819580291334317990998484185, −8.404871596259288604045417825207, −7.84298405200365218042406102391, −6.97939045327443668455989999066, −6.39601888460097042814218355910, −5.40272913042532002594487859668, −4.11636170255038719593876476212, −2.62223351766023596639712071955, −0.925409949379851576213774966540, 0,
0.925409949379851576213774966540, 2.62223351766023596639712071955, 4.11636170255038719593876476212, 5.40272913042532002594487859668, 6.39601888460097042814218355910, 6.97939045327443668455989999066, 7.84298405200365218042406102391, 8.404871596259288604045417825207, 9.513819580291334317990998484185
