L(s) = 1 | − 0.933i·3-s + (0.723 + 4.94i)5-s + 6.91·7-s + 8.12·9-s − 11.8·11-s − 13.4·13-s + (4.61 − 0.675i)15-s − 28.7i·17-s − 30.1·19-s − 6.45i·21-s − 39.5·23-s + (−23.9 + 7.15i)25-s − 15.9i·27-s − 18.9i·29-s − 6.41i·31-s + ⋯ |
L(s) = 1 | − 0.311i·3-s + (0.144 + 0.989i)5-s + 0.987·7-s + 0.903·9-s − 1.07·11-s − 1.03·13-s + (0.307 − 0.0450i)15-s − 1.69i·17-s − 1.58·19-s − 0.307i·21-s − 1.71·23-s + (−0.958 + 0.286i)25-s − 0.592i·27-s − 0.655i·29-s − 0.206i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8373729906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8373729906\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.723 - 4.94i)T \) |
good | 3 | \( 1 + 0.933iT - 9T^{2} \) |
| 7 | \( 1 - 6.91T + 49T^{2} \) |
| 11 | \( 1 + 11.8T + 121T^{2} \) |
| 13 | \( 1 + 13.4T + 169T^{2} \) |
| 17 | \( 1 + 28.7iT - 289T^{2} \) |
| 19 | \( 1 + 30.1T + 361T^{2} \) |
| 23 | \( 1 + 39.5T + 529T^{2} \) |
| 29 | \( 1 + 18.9iT - 841T^{2} \) |
| 31 | \( 1 + 6.41iT - 961T^{2} \) |
| 37 | \( 1 - 51.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 58.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 48.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 4.02T + 2.80e3T^{2} \) |
| 59 | \( 1 + 45.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.82iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 119. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 83.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 54.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 51.8iT - 9.40e3T^{2} \) |
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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
−9.374493696329922434059406200211, −7.926965134644083666261819790126, −7.69993319976908789889367587742, −6.86962454414231358522719385203, −5.92388221228532291171373689728, −4.86172331020060252045606736264, −4.12053940011222439541399582939, −2.50331446824106560905932430568, −2.11276465359009089658247279943, −0.22284368446484471498303684479,
1.49050897929468498429345583587, 2.30155236751358874830013476717, 4.20887960437023787805988284115, 4.44480660513515522385051837897, 5.43718161288390171895548387430, 6.29377448250488075268340677975, 7.72287160794371586828903243258, 8.039515938799780699211406884646, 8.870403441979924648738006432491, 9.927597205259006068913573861712