L(s) = 1 | + (−1.22 + 1.22i)3-s + (−3.32 + 3.32i)5-s − 4.04·7-s − 2.99i·9-s + (−6.82 − 6.82i)11-s + (−4.29 − 4.29i)13-s − 8.14i·15-s + 30.1·17-s + (19.7 − 19.7i)19-s + (4.94 − 4.94i)21-s − 28.2·23-s + 2.86i·25-s + (3.67 + 3.67i)27-s + (21.3 + 21.3i)29-s − 38.0i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.665 + 0.665i)5-s − 0.577·7-s − 0.333i·9-s + (−0.620 − 0.620i)11-s + (−0.330 − 0.330i)13-s − 0.543i·15-s + 1.77·17-s + (1.03 − 1.03i)19-s + (0.235 − 0.235i)21-s − 1.22·23-s + 0.114i·25-s + (0.136 + 0.136i)27-s + (0.736 + 0.736i)29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.651900 - 0.406612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.651900 - 0.406612i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
good | 5 | \( 1 + (3.32 - 3.32i)T - 25iT^{2} \) |
| 7 | \( 1 + 4.04T + 49T^{2} \) |
| 11 | \( 1 + (6.82 + 6.82i)T + 121iT^{2} \) |
| 13 | \( 1 + (4.29 + 4.29i)T + 169iT^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 + (-19.7 + 19.7i)T - 361iT^{2} \) |
| 23 | \( 1 + 28.2T + 529T^{2} \) |
| 29 | \( 1 + (-21.3 - 21.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + (-42.8 + 42.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (32.6 + 32.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 15.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-0.476 + 0.476i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (9.97 + 9.97i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (37.9 + 37.9i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (20.0 - 20.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 40.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 30.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 130. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-2.26 + 2.26i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 72.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112.T + 9.40e3T^{2} \) |
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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
−10.93910426238871212771491111008, −10.13965811375034492559082927057, −9.370347325130695753685012867212, −7.948833514814111541122058251656, −7.30966825392934412988434897605, −6.03972856051002819096608328519, −5.19421550049510685529757677071, −3.69893352614615021226328209399, −2.92461643739735859205707570619, −0.40412861978402106837040089947,
1.24591154495586678913752955909, 3.08019644782541679824850538373, 4.44615589269296858794040512335, 5.46117163732130582058679500570, 6.51562315415582614488824577923, 7.81779131249690839546085526938, 8.082470460683749508106941666490, 9.814270716992870063264542868857, 10.06893347757862927679808072237, 11.68903998119999375257340065226