L(s) = 1 | + 3.81i·2-s + 1.73i·3-s − 10.5·4-s + 6.86i·5-s − 6.60·6-s − 3.56i·7-s − 24.8i·8-s − 2.99·9-s − 26.1·10-s − 4.73i·11-s − 18.2i·12-s + 20.4i·13-s + 13.6·14-s − 11.8·15-s + 52.7·16-s + 20.9·17-s + ⋯ |
L(s) = 1 | + 1.90i·2-s + 0.577i·3-s − 2.63·4-s + 1.37i·5-s − 1.10·6-s − 0.509i·7-s − 3.11i·8-s − 0.333·9-s − 2.61·10-s − 0.430i·11-s − 1.51i·12-s + 1.56i·13-s + 0.971·14-s − 0.793·15-s + 3.29·16-s + 1.23·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.683844 - 0.652843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.683844 - 0.652843i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 67 | \( 1 + (66.9 - 3.10i)T \) |
good | 2 | \( 1 - 3.81iT - 4T^{2} \) |
| 5 | \( 1 - 6.86iT - 25T^{2} \) |
| 7 | \( 1 + 3.56iT - 49T^{2} \) |
| 11 | \( 1 + 4.73iT - 121T^{2} \) |
| 13 | \( 1 - 20.4iT - 169T^{2} \) |
| 17 | \( 1 - 20.9T + 289T^{2} \) |
| 19 | \( 1 + 13.7T + 361T^{2} \) |
| 23 | \( 1 + 26.1T + 529T^{2} \) |
| 29 | \( 1 - 47.0T + 841T^{2} \) |
| 31 | \( 1 - 1.48iT - 961T^{2} \) |
| 37 | \( 1 + 52.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 17.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 75.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 97.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 15.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.9iT - 3.72e3T^{2} \) |
| 71 | \( 1 - 4.45T + 5.04e3T^{2} \) |
| 73 | \( 1 + 3.98T + 5.32e3T^{2} \) |
| 79 | \( 1 - 55.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 99.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 91.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 36.9iT - 9.40e3T^{2} \) |
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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
−13.84382719570614200297772404940, −12.06380560340418528125467007033, −10.60294556928041480757509696423, −9.846721450442995390758316148239, −8.715317920096164620255864458539, −7.66994909413361793945522302517, −6.70641827717536422439700195410, −6.08256565386164790658863870255, −4.61432593120251070569921538173, −3.55879015553616924202853209216,
0.56650302045876479141048225923, 1.82352016202242235073187332920, 3.25604613028493674057147049643, 4.75072372672621075491060113286, 5.61819995191351666509563414828, 8.137019201032430801677810907361, 8.563507926803705033238778587559, 9.798851424501917505118924806025, 10.47960048609642197683258997423, 12.02011274295899749044253430077