L(s) = 1 | + (−0.392 + 0.680i)2-s + (−3.69 + 3.65i)3-s + (3.69 + 6.39i)4-s + (−1.03 − 3.94i)6-s + (10.4 − 18.1i)7-s − 12.0·8-s + (0.319 − 26.9i)9-s + (29.6 − 51.4i)11-s + (−36.9 − 10.1i)12-s + (−22.9 − 39.7i)13-s + (8.21 + 14.2i)14-s + (−24.7 + 42.9i)16-s − 43.0·17-s + (18.2 + 10.8i)18-s + 140.·19-s + ⋯ |
L(s) = 1 | + (−0.138 + 0.240i)2-s + (−0.711 + 0.702i)3-s + (0.461 + 0.799i)4-s + (−0.0702 − 0.268i)6-s + (0.564 − 0.977i)7-s − 0.533·8-s + (0.0118 − 0.999i)9-s + (0.813 − 1.40i)11-s + (−0.890 − 0.244i)12-s + (−0.489 − 0.848i)13-s + (0.156 + 0.271i)14-s + (−0.387 + 0.670i)16-s − 0.614·17-s + (0.238 + 0.141i)18-s + 1.69·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.34081 - 0.125308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34081 - 0.125308i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.69 - 3.65i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.392 - 0.680i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-10.4 + 18.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-29.6 + 51.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (22.9 + 39.7i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 43.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (50.8 + 88.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (6.19 - 10.7i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (35.9 + 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (25.5 + 44.2i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (17.2 - 29.8i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-58.9 + 102. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (248. + 429. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-123. + 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-413. - 715. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 260.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 372.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (238. - 412. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-156. + 271. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 817.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-470. + 815. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.45891670370038623297109691592, −11.08180478548973148562526665516, −9.936052653130396655250292076588, −8.748526211736000607369611777230, −7.72464527579264095080395361377, −6.68727026988365335876049019203, −5.59190195897914793731481730936, −4.19030815254810542195588883510, −3.20343001498158524675406702543, −0.68159319427352616619949085142,
1.43571505565665016997387872974, 2.25711383442695895625491728612, 4.74192519090869358238627252353, 5.64539884536606041286833601685, 6.73453891973906984855212176611, 7.52222504400995408662207423422, 9.159451783687104228220573941541, 9.866817795302165210089861462566, 11.19242646103060366034206033533, 11.87842164611412511855424984819