L(s) = 1 | − 1.73i·5-s + 3i·7-s − 5.19·11-s − 13-s + 3.46i·17-s + 6i·19-s − 5.19·23-s + 2.00·25-s − 8.66i·29-s − 3i·31-s + 5.19·35-s + 4·37-s + 5.19i·41-s − 3i·43-s − 5.19·47-s + ⋯ |
L(s) = 1 | − 0.774i·5-s + 1.13i·7-s − 1.56·11-s − 0.277·13-s + 0.840i·17-s + 1.37i·19-s − 1.08·23-s + 0.400·25-s − 1.60i·29-s − 0.538i·31-s + 0.878·35-s + 0.657·37-s + 0.811i·41-s − 0.457i·43-s − 0.757·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8676080178\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8676080178\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 8.66iT - 29T^{2} \) |
| 31 | \( 1 + 3iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 15iT - 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−8.185199207072621019515081043985, −7.66168289757404092506726796819, −6.28943504102682625826437351045, −5.83803446249892652618841646143, −5.19116596021185894088894066495, −4.46688717438742247857060477741, −3.49639214701660256852741795928, −2.41677449623091275869148967348, −1.84804952332720994998663885801, −0.26822038575757211468173403870,
0.904401654302912562778234477612, 2.42940190010054195981045922369, 2.92620695118381712624187567453, 3.86516692939501432175664717216, 4.85832008345101056193200477941, 5.28285344404272357640211333980, 6.47241956203452767829095717544, 7.08980893564509730246465390163, 7.48624289774944834344086339083, 8.210708548058941821138060096641