L(s) = 1 | + (−0.238 + 0.412i)2-s + (3.09 + 4.17i)3-s + (3.88 + 6.73i)4-s + (−2.45 + 0.279i)6-s + (−6.34 + 10.9i)7-s − 7.50·8-s + (−7.90 + 25.8i)9-s + (0.794 − 1.37i)11-s + (−16.1 + 37.0i)12-s + (−5.36 − 9.29i)13-s + (−3.01 − 5.22i)14-s + (−29.3 + 50.7i)16-s − 69.7·17-s + (−8.76 − 9.40i)18-s + 98.5·19-s + ⋯ |
L(s) = 1 | + (−0.0841 + 0.145i)2-s + (0.594 + 0.803i)3-s + (0.485 + 0.841i)4-s + (−0.167 + 0.0190i)6-s + (−0.342 + 0.592i)7-s − 0.331·8-s + (−0.292 + 0.956i)9-s + (0.0217 − 0.0377i)11-s + (−0.387 + 0.891i)12-s + (−0.114 − 0.198i)13-s + (−0.0576 − 0.0997i)14-s + (−0.457 + 0.793i)16-s − 0.995·17-s + (−0.114 − 0.123i)18-s + 1.19·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.435886 + 1.81431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.435886 + 1.81431i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.09 - 4.17i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.238 - 0.412i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (6.34 - 10.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-0.794 + 1.37i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (5.36 + 9.29i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-15.7 - 27.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-150. + 260. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-58.6 - 101. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (70.9 + 122. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (150. - 260. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (243. - 422. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 459.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-250. - 433. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (290. - 503. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (250. + 433. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 435.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (187. - 325. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-646. + 1.11e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 403.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-790. + 1.36e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.07529086169086914737189877330, −11.31184011250332338400675683610, −10.13925454627817114593519499179, −9.146465443908598191321060031061, −8.368523471857071646979196674393, −7.39218313435238997694450438905, −6.10266863169093142663070245789, −4.66477349966933203795601874461, −3.35297185462998795710358352621, −2.44439796770123840795325216447,
0.71255198024751090257940048072, 2.04305789009412974959206861849, 3.38698185216814078747576105564, 5.17876690288073989308095506432, 6.66665150179872637637584357475, 7.01895906931492066732095734426, 8.450494335661164080006451008817, 9.469251370983235058488764344327, 10.36681264562323196450566279426, 11.44533608568164318533412078027