| L(s) = 1 | + (−1.87 + 0.684i)2-s + (3.06 − 2.57i)4-s + (3.01 + 17.0i)5-s + (16.7 + 14.0i)7-s + (−4.00 + 6.92i)8-s + (−17.3 − 30.0i)10-s + (7.05 − 39.9i)11-s + (−46.7 − 17.0i)13-s + (−40.9 − 14.9i)14-s + (2.77 − 15.7i)16-s + (64.5 + 111. i)17-s + (−36.0 + 62.3i)19-s + (53.1 + 44.5i)20-s + (14.1 + 79.9i)22-s + (14.7 − 12.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.383 − 0.321i)4-s + (0.269 + 1.52i)5-s + (0.901 + 0.756i)7-s + (−0.176 + 0.306i)8-s + (−0.548 − 0.949i)10-s + (0.193 − 1.09i)11-s + (−0.997 − 0.362i)13-s + (−0.782 − 0.284i)14-s + (0.0434 − 0.246i)16-s + (0.920 + 1.59i)17-s + (−0.434 + 0.753i)19-s + (0.594 + 0.498i)20-s + (0.136 + 0.775i)22-s + (0.133 − 0.111i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.582231 + 1.04420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582231 + 1.04420i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.01 - 17.0i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-16.7 - 14.0i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-7.05 + 39.9i)T + (-1.25e3 - 455. i)T^{2} \) |
| 13 | \( 1 + (46.7 + 17.0i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-64.5 - 111. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (36.0 - 62.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-14.7 + 12.3i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (186. - 67.8i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (37.3 - 31.3i)T + (5.17e3 - 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-137. - 238. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (6.64 + 2.41i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-14.1 + 80.3i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-11.4 - 9.57i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (26.8 + 152. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (170. + 143. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-458. - 167. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-503. - 872. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-465. + 806. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-302. + 110. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-56.1 + 20.4i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-171. + 296. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-61.1 + 346. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−12.55372842047755445982421028208, −11.37873416993878476079308418984, −10.69975324283775238094059618549, −9.838794478681369877362403866570, −8.442740001953822300319591398663, −7.68618344555298257992583014921, −6.36810150841636332731191428900, −5.54250031360004508159133424288, −3.31440287070169636979532081561, −1.92191842481903571779020819029,
0.70324804469161431737981152163, 2.02497633698072728359415995198, 4.40264838002467377948072884751, 5.16626756188910600189629317734, 7.20831920333433874011919071391, 7.88614352005989091927382905395, 9.331238020997766338735560086886, 9.607131740014016140238305403132, 11.10107671293230945237323180496, 12.05098107732627820128391448793
