L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (4.24 + 2.64i)5-s + (8.20 − 4.73i)7-s + 2.82·8-s + (0.246 − 7.06i)10-s + (17.6 − 10.1i)11-s + (−5.29 − 3.05i)13-s + (−11.6 − 6.70i)14-s + (−2.00 − 3.46i)16-s − 24.0·17-s + 16.0·19-s + (−8.82 + 4.69i)20-s + (−24.9 − 14.4i)22-s + (21.6 − 37.4i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.848 + 0.529i)5-s + (1.17 − 0.676i)7-s + 0.353·8-s + (0.0246 − 0.706i)10-s + (1.60 − 0.926i)11-s + (−0.407 − 0.234i)13-s + (−0.828 − 0.478i)14-s + (−0.125 − 0.216i)16-s − 1.41·17-s + 0.846·19-s + (−0.441 + 0.234i)20-s + (−1.13 − 0.655i)22-s + (0.940 − 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.172937030\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.172937030\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.24 - 2.64i)T \) |
good | 7 | \( 1 + (-8.20 + 4.73i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5.29 + 3.05i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 24.0T + 289T^{2} \) |
| 19 | \( 1 - 16.0T + 361T^{2} \) |
| 23 | \( 1 + (-21.6 + 37.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (16.9 - 9.81i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (15.2 - 26.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 44.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (41.6 + 24.0i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-41.6 + 24.0i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (34.0 + 58.9i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 61.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-43.6 - 25.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-24.4 - 42.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (47.7 + 27.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 23.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (16.6 + 28.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (11.3 + 19.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 79.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (103. - 59.9i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.07750901761647545079104556902, −8.879471938655588436653039805843, −8.660282014034888396250404439092, −7.17149205302016686816312467213, −6.66736328114997997059614690107, −5.32536483151773032092505204406, −4.32984086253000967958300001599, −3.22018176308484855705891065758, −1.96529149802332438549865581886, −0.954955505428071032038466190752,
1.37443835009435485542209479990, 2.10867840810497040058141074356, 4.19214869429829118763110175521, 5.01720932569111207977287360609, 5.77697779453038769900922439740, 6.81503961604117005431523381688, 7.58944934130669573060684378268, 8.723466098105740517989722230383, 9.345516968497695001508507692555, 9.655852983647663071498294487658