L(s) = 1 | + (1.20 + 2.09i)2-s − 3·3-s + (1.08 − 1.87i)4-s − 7.44·5-s + (−3.62 − 6.27i)6-s + (−10.2 + 17.8i)7-s + 24.5·8-s + 9·9-s + (−8.98 − 15.5i)10-s + (−6.01 + 10.4i)11-s + (−3.24 + 5.62i)12-s + (−26.9 − 46.6i)13-s − 49.7·14-s + 22.3·15-s + (20.9 + 36.3i)16-s + (−37.4 − 64.9i)17-s + ⋯ |
L(s) = 1 | + (0.426 + 0.739i)2-s − 0.577·3-s + (0.135 − 0.234i)4-s − 0.665·5-s + (−0.246 − 0.426i)6-s + (−0.555 + 0.962i)7-s + 1.08·8-s + 0.333·9-s + (−0.284 − 0.492i)10-s + (−0.164 + 0.285i)11-s + (−0.0781 + 0.135i)12-s + (−0.574 − 0.994i)13-s − 0.949·14-s + 0.384·15-s + (0.327 + 0.567i)16-s + (−0.534 − 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.246570 - 0.342442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.246570 - 0.342442i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 67 | \( 1 + (446. - 319. i)T \) |
good | 2 | \( 1 + (-1.20 - 2.09i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 7.44T + 125T^{2} \) |
| 7 | \( 1 + (10.2 - 17.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (6.01 - 10.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (26.9 + 46.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (37.4 + 64.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (67.7 + 117. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (77.3 + 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (63.8 - 110. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (39.0 - 67.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-192. - 333. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-65.3 + 113. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-86.8 + 150. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 561.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 259.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (45.1 + 78.2i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 71 | \( 1 + (-250. + 433. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-92.7 - 160. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (263. - 455. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-58.8 - 101. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 903.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (183. + 317. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.80338462057871527173886204477, −10.81220157369498916381637002496, −9.837220315354093788174127032571, −8.495307908619862548898613235067, −7.25504592751294728658274433513, −6.46695198015026459644355820022, −5.37146566876802248723427848596, −4.52673421937929189079806977456, −2.55870140634317827911425108706, −0.15951962936369713239433264540,
1.87822142839203509563491904117, 3.85025016885562706497404826263, 4.14379916025563495639724541170, 6.01723337124611711283114669720, 7.23993618407334830468018816224, 8.013716582068653914864848037277, 9.758184416987994311928312637538, 10.62919199593421552060101275973, 11.45542208464756409206840836374, 12.14719477989156529400348612401