L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (−0.661 − 1.14i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−3.99 − 2.30i)11-s + 0.999i·12-s + (1.66 + 3.20i)13-s + 1.32·14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s − 0.999·18-s + (−1.98 + 1.14i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.249 − 0.432i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−1.20 − 0.695i)11-s + 0.288i·12-s + (0.460 + 0.887i)13-s + 0.353·14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s − 0.235·18-s + (−0.455 + 0.262i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3739225468\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3739225468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.66 - 3.20i)T \) |
good | 7 | \( 1 + (0.661 + 1.14i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.99 + 2.30i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.98 - 1.14i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.50 - 4.33i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.01 - 1.75i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 37 | \( 1 + (-3.40 + 5.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.02 + 2.32i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.45 - 4.30i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 - 0.826iT - 53T^{2} \) |
| 59 | \( 1 + (2.72 - 1.57i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.267 + 0.464i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.59 - 2.75i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.81 - 5.66i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + (10.2 + 5.93i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.40 - 7.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.820231617553510705972665693590, −7.967189393578359726914607642724, −7.34566455347099416930804643679, −6.63351865641185619637318729618, −5.73531892447573890964180769902, −5.19591312225430681613062139106, −4.08347845625867929469574422137, −2.93999965875440290276380987972, −1.45899157530161177186398045257, −0.17791436845095923230113868851,
1.32110351723315694892945962564, 2.73856732174122696160938306624, 3.35684352014078973626308151218, 4.75615832446305458515610818094, 5.19880620579894216389256626823, 6.26809428502783370888027964482, 7.15167372123432376308186707414, 8.153635029331310719939125498586, 8.664394194955672706045876008261, 9.674604047634895094257591339004