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
\(L(1)\) |
\(\approx\) |
\(0.271847 - 0.846481i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.271847 - 0.846481i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-6.18 + 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.349469557683545294059063815773, −8.680656115512182978637951982617, −7.78356114546115909749387633567, −7.46026878787887175793280871842, −6.45199690042714598160619331877, −5.79308865880552789027749101841, −4.39772317323798831103550129264, −2.87326198509076201944183694894, −2.02616284688674141097400409095, −0.39178078556806443530403983019,
2.10065824165310322638048762937, 3.18608928333135925943176363564, 3.83134659380687881883594045818, 4.84770694134586938464103069102, 5.79651999783414845454189514630, 7.37343232906486561212304468489, 8.221303980897565124143185909952, 8.898447061816280094900084040691, 9.675368408455805068303892548905, 10.28690726397963071994259744975