| L(s) = 1 | − 3.03·3-s + 2.23i·5-s + 6.21i·7-s + 0.181·9-s − 20.6i·11-s − 10.8·13-s − 6.77i·15-s + 10.5i·17-s − 8.71i·19-s − 18.8i·21-s + (21.1 − 9.04i)23-s − 5.00·25-s + 26.7·27-s + 5.68·29-s + 16.3·31-s + ⋯ |
| L(s) = 1 | − 1.01·3-s + 0.447i·5-s + 0.887i·7-s + 0.0201·9-s − 1.87i·11-s − 0.837·13-s − 0.451i·15-s + 0.621i·17-s − 0.458i·19-s − 0.896i·21-s + (0.919 − 0.393i)23-s − 0.200·25-s + 0.989·27-s + 0.196·29-s + 0.527·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6398879615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6398879615\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-21.1 + 9.04i)T \) |
| good | 3 | \( 1 + 3.03T + 9T^{2} \) |
| 7 | \( 1 - 6.21iT - 49T^{2} \) |
| 11 | \( 1 + 20.6iT - 121T^{2} \) |
| 13 | \( 1 + 10.8T + 169T^{2} \) |
| 17 | \( 1 - 10.5iT - 289T^{2} \) |
| 19 | \( 1 + 8.71iT - 361T^{2} \) |
| 29 | \( 1 - 5.68T + 841T^{2} \) |
| 31 | \( 1 - 16.3T + 961T^{2} \) |
| 37 | \( 1 - 34.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 45.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 21.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 28.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 62.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 92.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 60.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 6.11iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 129.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 94.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 10.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 54.0iT - 9.40e3T^{2} \) |
| 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
−9.178186353393677147507151393413, −8.620162589095097074366702356387, −7.76909935328201834209899414985, −6.58700174627850329929818729130, −6.10728778110089417434667791786, −5.44889247935094868603804578412, −4.64730845765666284962131341363, −3.22132656126112705707455910744, −2.58238505937292023987137006925, −0.893049600095323922864542522991,
0.25209602387628060808511862752, 1.41785016526380976444143532000, 2.69979580659556184260922211432, 4.18532675955171647434364479007, 4.77824761122834546746026618867, 5.40334119811959488867608859783, 6.50321638123671402812360560606, 7.25349739371684622348600625702, 7.70800459453215644914054123778, 8.995967479867062915199302487275
