L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.23i·5-s + 3.95i·7-s − 2.82·8-s − 3.16i·10-s + 5.67i·11-s + 11.1·13-s − 5.59i·14-s + 4.00·16-s − 2.12i·17-s − 20.9i·19-s + 4.47i·20-s − 8.02i·22-s + (10.6 − 20.3i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.447i·5-s + 0.565i·7-s − 0.353·8-s − 0.316i·10-s + 0.515i·11-s + 0.856·13-s − 0.399i·14-s + 0.250·16-s − 0.125i·17-s − 1.10i·19-s + 0.223i·20-s − 0.364i·22-s + (0.464 − 0.885i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.302778641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.302778641\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-10.6 + 20.3i)T \) |
good | 7 | \( 1 - 3.95iT - 49T^{2} \) |
| 11 | \( 1 - 5.67iT - 121T^{2} \) |
| 13 | \( 1 - 11.1T + 169T^{2} \) |
| 17 | \( 1 + 2.12iT - 289T^{2} \) |
| 19 | \( 1 + 20.9iT - 361T^{2} \) |
| 29 | \( 1 + 36.0T + 841T^{2} \) |
| 31 | \( 1 - 39.7T + 961T^{2} \) |
| 37 | \( 1 + 50.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 60.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.29iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 83.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 61.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 0.572T + 5.04e3T^{2} \) |
| 73 | \( 1 - 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 86.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 84.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 94.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 171. iT - 9.40e3T^{2} \) |
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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
−8.832558760300541000265057755340, −8.307615243209462012526860579098, −7.24613051947317244840193264049, −6.73777830876335902441779308658, −5.87242167055725750855912211399, −4.95097802047221173821464809981, −3.79085241229513228172316030999, −2.73958134204737933050320874181, −1.92926382355748256779748883446, −0.52060053862132319838908932415,
0.918202018112244759461603517398, 1.68715298994517947370696629629, 3.18917935316330681561640483851, 3.90739463914377278271588193925, 5.08403784988130516409712297103, 5.99114828672754283897594686494, 6.68222620482233248621680954985, 7.70099109546971528911423581318, 8.251197745193191863136983800268, 8.889733553456967100618323338872