| L(s) = 1 | + (3 − 3i)3-s + (42.8 + 42.8i)7-s + 63i·9-s − 181.·11-s + (−25.1 + 25.1i)13-s + (−160. − 160. i)17-s + 80.2i·19-s + 257.·21-s + (−324. + 324. i)23-s + (432 + 432i)27-s + 1.37e3i·29-s − 151.·31-s + (−545. + 545. i)33-s + (1.42e3 + 1.42e3i)37-s + 150. i·39-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.333i)3-s + (0.874 + 0.874i)7-s + 0.777i·9-s − 1.50·11-s + (−0.148 + 0.148i)13-s + (−0.555 − 0.555i)17-s + 0.222i·19-s + 0.583·21-s + (−0.613 + 0.613i)23-s + (0.592 + 0.592i)27-s + 1.63i·29-s − 0.157·31-s + (−0.500 + 0.500i)33-s + (1.04 + 1.04i)37-s + 0.0992i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.438218833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438218833\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-3 + 3i)T - 81iT^{2} \) |
| 7 | \( 1 + (-42.8 - 42.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 181.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (25.1 - 25.1i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (160. + 160. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 80.2iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (324. - 324. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 1.37e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 151.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.42e3 - 1.42e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 3.01e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.61e3 + 1.61e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-397. - 397. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (923. - 923. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 4.54e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 631.T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-2.68e3 - 2.68e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 4.54e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (5.85e3 - 5.85e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 9.42e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-8.39e3 + 8.39e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 8.36e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.01e3 + 1.01e3i)T + 8.85e7iT^{2} \) |
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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
−12.07687947622378540720028406814, −11.11744571125405199605352324868, −10.21376858691975925274165323556, −8.819643624429639821938974418892, −8.075542966475353554406489967132, −7.22267038856715602964640693302, −5.53475547333505282182806963261, −4.82249340129072188896161908269, −2.80370490822258761139364600802, −1.82681275201194259259019278560,
0.46187094050127873801729262286, 2.33920060926883574627003423580, 3.88065279264343908263429237676, 4.87126972656200099504434797386, 6.29369626084751354354739535337, 7.68013168951677639421964802223, 8.322277607744480621689023344041, 9.659598357452507196327398703784, 10.51004168328682600026441304060, 11.31933351950703979539687208660
