| L(s) = 1 | − 1.89·3-s + 0.513·7-s + 0.591·9-s + 1.48·11-s − 5.38·13-s + 17-s + 2.40·19-s − 0.973·21-s − 0.513·23-s + 4.56·27-s + 4.40·29-s − 7.68·31-s − 2.81·33-s − 9.79·37-s + 10.1·39-s + 4.97·41-s + 2·43-s + 0.408·47-s − 6.73·49-s − 1.89·51-s − 10.9·53-s − 4.56·57-s + 12.3·59-s + 0.618·61-s + 0.303·63-s + 14.6·67-s + 0.973·69-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 0.194·7-s + 0.197·9-s + 0.448·11-s − 1.49·13-s + 0.242·17-s + 0.552·19-s − 0.212·21-s − 0.107·23-s + 0.878·27-s + 0.818·29-s − 1.38·31-s − 0.490·33-s − 1.60·37-s + 1.63·39-s + 0.776·41-s + 0.304·43-s + 0.0595·47-s − 0.962·49-s − 0.265·51-s − 1.50·53-s − 0.604·57-s + 1.60·59-s + 0.0791·61-s + 0.0382·63-s + 1.78·67-s + 0.117·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8901257505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8901257505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 7 | \( 1 - 0.513T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 5.38T + 13T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 + 0.513T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 9.79T + 37T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 0.408T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 0.618T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6.65T + 71T^{2} \) |
| 73 | \( 1 + 0.408T + 73T^{2} \) |
| 79 | \( 1 - 3.54T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 9.98T + 89T^{2} \) |
| 97 | \( 1 + 8.61T + 97T^{2} \) |
| 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
−7.85767445741910680714640284942, −7.08405180890120231867986907544, −6.64210429604199754479104063219, −5.70469924563449377710043500835, −5.20420798114090909398236412512, −4.65325785375873072214116044781, −3.65710129827447208880228591075, −2.71078754488980848757120429151, −1.67465608981473820330801436535, −0.51108979937326385821119212461,
0.51108979937326385821119212461, 1.67465608981473820330801436535, 2.71078754488980848757120429151, 3.65710129827447208880228591075, 4.65325785375873072214116044781, 5.20420798114090909398236412512, 5.70469924563449377710043500835, 6.64210429604199754479104063219, 7.08405180890120231867986907544, 7.85767445741910680714640284942
