| L(s) = 1 | − 0.558i·5-s − 0.558i·7-s − 11-s − 1.30·13-s + 5.94i·17-s + 6.72i·19-s − 4.99·23-s + 4.68·25-s + 3.46i·29-s − 2.48i·31-s − 0.311·35-s + 9.62·37-s − 2.48i·41-s + 2.87i·43-s + 6.92·47-s + ⋯ |
| L(s) = 1 | − 0.249i·5-s − 0.210i·7-s − 0.301·11-s − 0.362·13-s + 1.44i·17-s + 1.54i·19-s − 1.04·23-s + 0.937·25-s + 0.643i·29-s − 0.446i·31-s − 0.0526·35-s + 1.58·37-s − 0.387i·41-s + 0.438i·43-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.349804916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349804916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 0.558iT - 5T^{2} \) |
| 7 | \( 1 + 0.558iT - 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 5.94iT - 17T^{2} \) |
| 19 | \( 1 - 6.72iT - 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 2.48iT - 31T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 + 2.48iT - 41T^{2} \) |
| 43 | \( 1 - 2.87iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 7.48iT - 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 - 2.69T + 61T^{2} \) |
| 67 | \( 1 - 5.02iT - 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 1.56iT - 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
| 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
−9.650491448548357379133758052167, −8.647124337606652824040142077774, −8.018357196574461476962054771900, −7.29982378734631135021708282753, −6.14294749084033632659280000393, −5.63942012436915772082095001642, −4.41329579502194671384424231712, −3.76101295990158434789563653809, −2.46662866373365131129706620299, −1.27990723831432594177750850224,
0.55999289882394860601571794481, 2.35264893764879901330680475324, 2.98875909459224376187358619572, 4.37237914930220020920879784318, 5.06719276926293711490000656742, 6.03502097554855875893681106241, 6.99182855050615344184098118954, 7.53597123480060435790859359725, 8.557736658446712224358603248288, 9.311349631423405298678145928204
