| L(s) = 1 | − 5.99·3-s + 5·5-s − 6.65·7-s + 26.8·9-s − 29.9·15-s + 39.8·21-s + 40.5·23-s + 25·25-s − 107.·27-s − 53.6·29-s − 33.2·35-s − 53.6·41-s − 25.2·43-s + 134.·45-s + 69.2·47-s − 4.66·49-s − 58·61-s − 179.·63-s + 34.5·67-s − 243.·69-s − 149.·75-s + 399.·81-s + 50.6·83-s + 321.·87-s + 142·89-s + 160.·101-s − 111.·103-s + ⋯ |
| L(s) = 1 | − 1.99·3-s + 5-s − 0.951·7-s + 2.98·9-s − 1.99·15-s + 1.89·21-s + 1.76·23-s + 25-s − 3.96·27-s − 1.85·29-s − 0.951·35-s − 1.30·41-s − 0.587·43-s + 2.98·45-s + 1.47·47-s − 0.0952·49-s − 0.950·61-s − 2.84·63-s + 0.516·67-s − 3.52·69-s − 1.99·75-s + 4.93·81-s + 0.609·83-s + 3.69·87-s + 1.59·89-s + 1.59·101-s − 1.07·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9144593885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9144593885\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 3 | \( 1 + 5.99T + 9T^{2} \) |
| 7 | \( 1 + 6.65T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 40.5T + 529T^{2} \) |
| 29 | \( 1 + 53.6T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 53.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 58T + 3.72e3T^{2} \) |
| 67 | \( 1 - 34.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 142T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.661444733445550382887149982905, −9.088448190700450320037172266274, −7.34223898693914928053289275438, −6.76749300338934700607953930767, −6.07745907236809940032249462177, −5.42861795944238118415051606916, −4.74442266588613500733054018599, −3.44192736091668191855923241096, −1.80518375195142570530949667186, −0.63115392378885130257010954375,
0.63115392378885130257010954375, 1.80518375195142570530949667186, 3.44192736091668191855923241096, 4.74442266588613500733054018599, 5.42861795944238118415051606916, 6.07745907236809940032249462177, 6.76749300338934700607953930767, 7.34223898693914928053289275438, 9.088448190700450320037172266274, 9.661444733445550382887149982905
