| L(s) = 1 | + 1.73i·3-s + 7.85i·5-s − 2.99·9-s + 20.4·11-s − 4.69i·13-s − 13.6·15-s + 18.3i·17-s + 18.9i·19-s + 29.3·23-s − 36.7·25-s − 5.19i·27-s + 37.7·29-s + 34.0i·31-s + 35.4i·33-s + 42.8·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.57i·5-s − 0.333·9-s + 1.85·11-s − 0.360i·13-s − 0.907·15-s + 1.07i·17-s + 0.998i·19-s + 1.27·23-s − 1.47·25-s − 0.192i·27-s + 1.30·29-s + 1.09i·31-s + 1.07i·33-s + 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.150106067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150106067\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 7.85iT - 25T^{2} \) |
| 11 | \( 1 - 20.4T + 121T^{2} \) |
| 13 | \( 1 + 4.69iT - 169T^{2} \) |
| 17 | \( 1 - 18.3iT - 289T^{2} \) |
| 19 | \( 1 - 18.9iT - 361T^{2} \) |
| 23 | \( 1 - 29.3T + 529T^{2} \) |
| 29 | \( 1 - 37.7T + 841T^{2} \) |
| 31 | \( 1 - 34.0iT - 961T^{2} \) |
| 37 | \( 1 - 42.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.98iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 14.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 111.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 39.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 84.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 41.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 4.73iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138. iT - 9.40e3T^{2} \) |
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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
−10.05666097722942999883069888225, −9.153475971115355290134380092575, −8.329914746415783512301615578171, −7.23753970824149990512988532099, −6.46982454099706585064073972374, −5.95339446796510978533786606125, −4.50800959771782054740590870643, −3.59831723000659517252564106430, −2.96709364928589678941237504024, −1.46887200637471849398328805022,
0.74435256014288565457177013453, 1.37243911632947917520580428121, 2.84659339720978072977010572171, 4.34983502768012671311960709569, 4.78419143130002863606984696758, 5.99655014932686046765423025808, 6.77556186951873715647657282460, 7.63721101028627565703545181659, 8.740517918508185443548550484142, 9.098372988990162858486906419981
