| L(s) = 1 | + 4.68i·5-s + 3.30·7-s − 3.31i·11-s + 1.00·13-s − 6.45i·17-s − 25.1·19-s + 24.4i·23-s + 3.01·25-s + 50.8i·29-s − 48.4·31-s + 15.4i·35-s + 65.8·37-s − 6.41i·41-s + 48.5·43-s + 56.8i·47-s + ⋯ |
| L(s) = 1 | + 0.937i·5-s + 0.471·7-s − 0.301i·11-s + 0.0775·13-s − 0.379i·17-s − 1.32·19-s + 1.06i·23-s + 0.120·25-s + 1.75i·29-s − 1.56·31-s + 0.442i·35-s + 1.77·37-s − 0.156i·41-s + 1.12·43-s + 1.21i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.113540412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113540412\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 5 | \( 1 - 4.68iT - 25T^{2} \) |
| 7 | \( 1 - 3.30T + 49T^{2} \) |
| 13 | \( 1 - 1.00T + 169T^{2} \) |
| 17 | \( 1 + 6.45iT - 289T^{2} \) |
| 19 | \( 1 + 25.1T + 361T^{2} \) |
| 23 | \( 1 - 24.4iT - 529T^{2} \) |
| 29 | \( 1 - 50.8iT - 841T^{2} \) |
| 31 | \( 1 + 48.4T + 961T^{2} \) |
| 37 | \( 1 - 65.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.41iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 67.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.307iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 29.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 61.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 85.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 64.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 86.8T + 9.40e3T^{2} \) |
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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
−9.467576846270366304993673787827, −8.829377573754687033081815925455, −7.82605653771644371618975409207, −7.19797807879019958458133136766, −6.36629093614198184150985358437, −5.54271056521118346406607017761, −4.52421154494856136701359710952, −3.50512259947264410110579554373, −2.62206902363993402314182422497, −1.44832822182842890221239541077,
0.29563649342085073276951884752, 1.58429762556517384645368579402, 2.59037259014859337208082298104, 4.23092235061857356508442057280, 4.47066646249054994153187015297, 5.65608972882102770542145484117, 6.34621117991123380948605427204, 7.48608694903165961389899801824, 8.216175581374366298713334269148, 8.845744268066386429994685240982
