| L(s) = 1 | + 0.859i·2-s + 3.26·4-s − 6.29·7-s + 6.23i·8-s + 7.37i·11-s − 3.27·13-s − 5.41i·14-s + 7.68·16-s + 6.55i·17-s + 17.7·19-s − 6.33·22-s + 0.135i·23-s − 2.81i·26-s − 20.5·28-s + 21.0i·29-s + ⋯ |
| L(s) = 1 | + 0.429i·2-s + 0.815·4-s − 0.899·7-s + 0.779i·8-s + 0.670i·11-s − 0.251·13-s − 0.386i·14-s + 0.480·16-s + 0.385i·17-s + 0.935·19-s − 0.288·22-s + 0.00587i·23-s − 0.108i·26-s − 0.733·28-s + 0.726i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.445615267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445615267\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.859iT - 4T^{2} \) |
| 7 | \( 1 + 6.29T + 49T^{2} \) |
| 11 | \( 1 - 7.37iT - 121T^{2} \) |
| 13 | \( 1 + 3.27T + 169T^{2} \) |
| 17 | \( 1 - 6.55iT - 289T^{2} \) |
| 19 | \( 1 - 17.7T + 361T^{2} \) |
| 23 | \( 1 - 0.135iT - 529T^{2} \) |
| 29 | \( 1 - 21.0iT - 841T^{2} \) |
| 31 | \( 1 + 18.0T + 961T^{2} \) |
| 37 | \( 1 + 32.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 78.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 23.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 80.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 24.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 83.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 7.09T + 6.24e3T^{2} \) |
| 83 | \( 1 - 132. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 48.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 49.6T + 9.40e3T^{2} \) |
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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
−10.01859046334043886447586064681, −9.151508827624505530235989111018, −8.184722398417502059174873996157, −7.18893795642812269941218985849, −6.84464958140375992795305247915, −5.83259729191958830430994690434, −5.05915517070011233642144045731, −3.65857199050935967087264752475, −2.74797839079354140777857853904, −1.55561081833828486417205499200,
0.40093266546127783328555276817, 1.82600864360353992130642069686, 3.06935804274964598752231099550, 3.53276504533931378605663498572, 5.04511014641323927783171381535, 6.06521624330405542617465209091, 6.75266715272772579673574364649, 7.53345217763533113922205222814, 8.515536565262959500344276549915, 9.690330350293082545165657140393
