L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.39 − 2.41i)3-s + (−0.499 + 0.866i)4-s + (1.39 − 2.41i)6-s + 7-s − 0.999·8-s + (−2.39 + 4.14i)9-s − 3.79·11-s + 2.79·12-s + (−0.104 + 0.180i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.395 − 0.685i)17-s − 4.79·18-s + (−3.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.805 − 1.39i)3-s + (−0.249 + 0.433i)4-s + (0.569 − 0.986i)6-s + 0.377·7-s − 0.353·8-s + (−0.798 + 1.38i)9-s − 1.14·11-s + 0.805·12-s + (−0.0289 + 0.0501i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.0959 − 0.166i)17-s − 1.12·18-s + (−0.802 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0708259 + 0.220538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0708259 + 0.220538i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 3 | \( 1 + (1.39 + 2.41i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + (0.104 - 0.180i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.395 + 0.685i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.29 + 3.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.39 - 5.88i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + (-5.68 - 9.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.89 - 5.01i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.29 - 3.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.29 + 3.96i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.686 - 1.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 + 13.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 3.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.39 - 2.41i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.47 + 12.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + (2.29 - 3.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.18 + 10.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.79768733659773530509477272183, −9.378437223703305077800482018453, −8.151757904243355665385491907721, −7.78933389533083435830593608459, −6.87732663761693864358848991356, −6.19375803704122686745697877803, −5.37474007567763474102953803209, −4.57652121498817056118746524337, −2.89791544025812513703614700364, −1.61243259455048431009984848406,
0.10217185655563589183540727933, 2.23301595935747769826021735541, 3.57416520509140839848254950579, 4.34493216790591419390008030818, 5.30827631537671279337975948631, 5.61871130393530834677856707584, 7.01675482629958724454542097448, 8.279589469449587235389293294450, 9.240458781463918799138786150700, 9.949154829370621062128039307794