L(s) = 1 | + (1.5 + 0.866i)5-s + (−2.59 + 1.5i)7-s + (−2.59 − 4.5i)11-s + (0.5 − 0.866i)13-s + 3.46i·17-s − 6i·19-s + (−2.59 + 4.5i)23-s + (−1 − 1.73i)25-s + (−7.5 + 4.33i)29-s + (−2.59 − 1.5i)31-s − 5.19·35-s + 4·37-s + (−4.5 − 2.59i)41-s + (2.59 − 1.5i)43-s + (−2.59 − 4.5i)47-s + ⋯ |
L(s) = 1 | + (0.670 + 0.387i)5-s + (−0.981 + 0.566i)7-s + (−0.783 − 1.35i)11-s + (0.138 − 0.240i)13-s + 0.840i·17-s − 1.37i·19-s + (−0.541 + 0.938i)23-s + (−0.200 − 0.346i)25-s + (−1.39 + 0.804i)29-s + (−0.466 − 0.269i)31-s − 0.878·35-s + 0.657·37-s + (−0.702 − 0.405i)41-s + (0.396 − 0.228i)43-s + (−0.378 − 0.656i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4278741457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4278741457\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 + 4.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (2.59 - 4.5i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.5 - 4.33i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.59 + 1.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (-12.9 + 7.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.108134110181571077125906715959, −8.332601484331397124209054012995, −7.40052733898154371876692769283, −6.36959212202039799657252389758, −5.87578892799497170027111531797, −5.20613454816574619967985868497, −3.65124903512861400395034632662, −3.00321029404971547304909785928, −2.00282126828077988919770977002, −0.14772359698761493563895235260,
1.60157070132538580026666493638, 2.62067104253014564276148319392, 3.83197226639715680106094606029, 4.67251176138966250500559075600, 5.66133681573721058407295119183, 6.36010473817529224730006033325, 7.34420949404539865207900061506, 7.85074029452156713323245054689, 9.109973319745137435407710846236, 9.741037317772591818819528896680