| L(s) = 1 | + 2.84·3-s + 2.23·5-s + 3.16i·7-s − 0.887·9-s + 1.15i·13-s + 6.36·15-s + 17.5i·17-s + 20.0i·19-s + 9.01i·21-s − 36.9·23-s + 5.00·25-s − 28.1·27-s − 5.00i·29-s − 61.1·31-s + 7.07i·35-s + ⋯ |
| L(s) = 1 | + 0.949·3-s + 0.447·5-s + 0.452i·7-s − 0.0986·9-s + 0.0885i·13-s + 0.424·15-s + 1.03i·17-s + 1.05i·19-s + 0.429i·21-s − 1.60·23-s + 0.200·25-s − 1.04·27-s − 0.172i·29-s − 1.97·31-s + 0.202i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7795657989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7795657989\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.84T + 9T^{2} \) |
| 7 | \( 1 - 3.16iT - 49T^{2} \) |
| 13 | \( 1 - 1.15iT - 169T^{2} \) |
| 17 | \( 1 - 17.5iT - 289T^{2} \) |
| 19 | \( 1 - 20.0iT - 361T^{2} \) |
| 23 | \( 1 + 36.9T + 529T^{2} \) |
| 29 | \( 1 + 5.00iT - 841T^{2} \) |
| 31 | \( 1 + 61.1T + 961T^{2} \) |
| 37 | \( 1 + 26.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 26.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.09T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.31T + 3.48e3T^{2} \) |
| 61 | \( 1 + 33.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 17.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 41.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 42.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 109.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 20.0T + 9.40e3T^{2} \) |
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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
−8.974571657695429616143013695560, −8.456423904528071816280871864479, −7.82178593848562173108549533592, −6.90816196526916888760088369351, −5.76271632955760923107715439983, −5.55479270422444914462375032901, −3.95060359714113281790156352560, −3.55037832603159004683057080147, −2.20905280310797240779454119822, −1.82739558288042118796951315780,
0.13848682864015244424811847369, 1.62940490322693700738289515821, 2.59435235954579377110927474556, 3.32497853482693905804391656900, 4.30325777054530748782259410704, 5.23760413010839323771538385755, 6.10658922275077174895258421732, 7.05705987651970233140495719258, 7.70805130390452841761376977300, 8.442387184077120257227159217054
