L(s) = 1 | + (3.24 − 3.24i)3-s + (0.0586 − 0.0586i)5-s + 4.61·7-s − 12.1i·9-s + (5.36 + 5.36i)11-s + (−11.0 − 11.0i)13-s − 0.381i·15-s − 12.8·17-s + (−2.63 + 2.63i)19-s + (14.9 − 14.9i)21-s + 16.3·23-s + 24.9i·25-s + (−10.1 − 10.1i)27-s + (26.0 + 26.0i)29-s − 20.2i·31-s + ⋯ |
L(s) = 1 | + (1.08 − 1.08i)3-s + (0.0117 − 0.0117i)5-s + 0.659·7-s − 1.34i·9-s + (0.487 + 0.487i)11-s + (−0.850 − 0.850i)13-s − 0.0254i·15-s − 0.757·17-s + (−0.138 + 0.138i)19-s + (0.714 − 0.714i)21-s + 0.712·23-s + 0.999i·25-s + (−0.374 − 0.374i)27-s + (0.898 + 0.898i)29-s − 0.652i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73984 - 0.939044i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73984 - 0.939044i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-3.24 + 3.24i)T - 9iT^{2} \) |
| 5 | \( 1 + (-0.0586 + 0.0586i)T - 25iT^{2} \) |
| 7 | \( 1 - 4.61T + 49T^{2} \) |
| 11 | \( 1 + (-5.36 - 5.36i)T + 121iT^{2} \) |
| 13 | \( 1 + (11.0 + 11.0i)T + 169iT^{2} \) |
| 17 | \( 1 + 12.8T + 289T^{2} \) |
| 19 | \( 1 + (2.63 - 2.63i)T - 361iT^{2} \) |
| 23 | \( 1 - 16.3T + 529T^{2} \) |
| 29 | \( 1 + (-26.0 - 26.0i)T + 841iT^{2} \) |
| 31 | \( 1 + 20.2iT - 961T^{2} \) |
| 37 | \( 1 + (41.2 - 41.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 3.29iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-0.786 - 0.786i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 79.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (1.06 - 1.06i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (32.5 + 32.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (15.2 + 15.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-60.0 + 60.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 56.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 9.70iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 84.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (26.7 - 26.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 146.T + 9.40e3T^{2} \) |
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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
−12.95429214085848473724747892674, −12.27048628079827329002978872918, −10.97945248040303315014544194260, −9.526152782318713627414331866161, −8.489766680867538082155554021381, −7.60847098300423229323306918598, −6.71156979975870505960367452964, −4.88065726127518732003413406652, −3.00825781343484003453562791147, −1.60864747469980930110838385224,
2.41126381654477452517035283395, 3.96611107506475560551157912150, 4.91742886698349205788753390764, 6.82492573142846914820080938931, 8.331001625975792076547357465157, 8.988756179862012596048183696797, 9.991481953202706505719990615525, 11.03471241025248525801854116364, 12.14315108553648799250052361182, 13.74634270537817178139017551748