| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 4.01i·5-s + 13.7·7-s − 2.82i·8-s − 5.67·10-s − 19.5i·11-s + 17.2·13-s + 19.5i·14-s + 4.00·16-s − 9.21i·17-s − 14.2·19-s − 8.02i·20-s + 27.5·22-s + 4.79i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.802i·5-s + 1.97·7-s − 0.353i·8-s − 0.567·10-s − 1.77i·11-s + 1.32·13-s + 1.39i·14-s + 0.250·16-s − 0.541i·17-s − 0.750·19-s − 0.401i·20-s + 1.25·22-s + 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85616 + 0.960819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85616 + 0.960819i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 5 | \( 1 - 4.01iT - 25T^{2} \) |
| 7 | \( 1 - 13.7T + 49T^{2} \) |
| 11 | \( 1 + 19.5iT - 121T^{2} \) |
| 13 | \( 1 - 17.2T + 169T^{2} \) |
| 17 | \( 1 + 9.21iT - 289T^{2} \) |
| 19 | \( 1 + 14.2T + 361T^{2} \) |
| 29 | \( 1 - 30.9iT - 841T^{2} \) |
| 31 | \( 1 + 0.461T + 961T^{2} \) |
| 37 | \( 1 - 40.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 66.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 15.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 100.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 150.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 58.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 15.3T + 9.40e3T^{2} \) |
| 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
−11.10062844170107074960520480873, −10.55671435098913269882153260581, −8.803644558986026041142185082040, −8.419615376214497523289273977851, −7.52253573843529797814166878471, −6.36235760183085826761162985338, −5.54393283114749912503349308521, −4.41242238596630002030123929016, −3.11649401531927287618213157393, −1.24038466799507152090801344477,
1.29732465134035573600361069860, 2.09353106195814548466899374457, 4.29546046367032993572831259725, 4.56387069773472129033606534027, 5.79983886466128733818428678573, 7.44606753330304814672651616457, 8.357158083945540284289432868675, 8.912047004963068349947973232264, 10.15446566651545081787110554063, 10.96487564585574437452705927689
