L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + 2i·7-s + 0.999i·8-s + (−1 + 1.73i)9-s − 0.999i·12-s + (−5.19 − 3i)13-s + (−1 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (−6.06 + 3.5i)17-s − 2i·18-s + (−3.5 − 2.59i)19-s + (1 + 1.73i)21-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + 0.755i·7-s + 0.353i·8-s + (−0.333 + 0.577i)9-s − 0.288i·12-s + (−1.44 − 0.832i)13-s + (−0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (−1.47 + 0.848i)17-s − 0.471i·18-s + (−0.802 − 0.596i)19-s + (0.218 + 0.377i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00961971 - 0.141422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00961971 - 0.141422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.5 + 2.59i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (5.19 + 3i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (6.06 - 3.5i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (10.3 - 6i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.59 - 1.5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13iT - 83T^{2} \) |
| 89 | \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.9 - 7.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.33548050944378842357514993675, −9.501974315146967823020482792092, −8.523909607709718904140546743619, −8.236339617869257148942786708471, −7.24903724510044528233062592458, −6.35787579211384244387026013447, −5.40417115485968476051934839357, −4.43416090502937850857314655864, −2.62391822363794850042053879788, −2.16012183911447001980077903740,
0.06793423767732642485539334538, 1.92439053178790940389031801423, 3.00779027933957581370270984600, 4.07785054821255127927936154037, 4.92186111585557371088124051254, 6.63303956935156635565896586996, 7.01161453847289977576748745334, 8.171966803483250100996685842634, 8.909704252620267434109523954483, 9.580685911752075489826085605444