L(s) = 1 | + (−1.03 − 1.79i)2-s + (1.85 − 3.20i)4-s + (−2.5 + 4.33i)5-s + (2.33 + 4.03i)7-s − 24.2·8-s + 10.3·10-s + (−4.44 − 7.70i)11-s + (−17.3 + 29.9i)13-s + (4.83 − 8.36i)14-s + (10.3 + 17.9i)16-s + 2.66·17-s + 125.·19-s + (9.25 + 16.0i)20-s + (−9.21 + 15.9i)22-s + (−65.9 + 114. i)23-s + ⋯ |
L(s) = 1 | + (−0.366 − 0.634i)2-s + (0.231 − 0.400i)4-s + (−0.223 + 0.387i)5-s + (0.125 + 0.217i)7-s − 1.07·8-s + 0.327·10-s + (−0.121 − 0.211i)11-s + (−0.369 + 0.639i)13-s + (0.0922 − 0.159i)14-s + (0.161 + 0.279i)16-s + 0.0380·17-s + 1.51·19-s + (0.103 + 0.179i)20-s + (−0.0893 + 0.154i)22-s + (−0.598 + 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.282985922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.282985922\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (1.03 + 1.79i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-2.33 - 4.03i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (4.44 + 7.70i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (17.3 - 29.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 2.66T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (65.9 - 114. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-35.6 - 61.6i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (6.71 - 11.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (191. - 332. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-169. - 293. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (39.1 + 67.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 254.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-16.4 + 28.4i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (93.4 + 161. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (203. - 352. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 276.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-573. - 993. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (89.0 + 154. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-619. - 1.07e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.99287423025038284778908381979, −9.835076019794852975101565557758, −9.454207672231262081064387853846, −8.176381758524050792674457766050, −7.14134006613753357172045694081, −6.09785381233313696305577344688, −5.08993953937233316520112055508, −3.51259685764783207292915434329, −2.43378700435334734170608993485, −1.12704334570821606898504555551,
0.55379698533793517296256809791, 2.53148285091781437053483612048, 3.79525189942613279740580099802, 5.12495904036175116898980973582, 6.18004546694946373878199711471, 7.36817024577935611218333458746, 7.84083169553229284120118280938, 8.797287799342636862666389681082, 9.714223600035032567548455725028, 10.75035021729526484457554241902