L(s) = 1 | − 1.41·2-s + (0.292 + 0.707i)3-s + 2.00·4-s + (−0.414 − 1.00i)6-s + (−1 + i)7-s − 2.82·8-s + (1.70 − 1.70i)9-s + (−4.12 − 1.70i)11-s + (0.585 + 1.41i)12-s + (0.707 − 0.292i)13-s + (1.41 − 1.41i)14-s + 4.00·16-s − 2.82·17-s + (−2.41 + 2.41i)18-s + (−1.53 − 3.70i)19-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (0.169 + 0.408i)3-s + 1.00·4-s + (−0.169 − 0.408i)6-s + (−0.377 + 0.377i)7-s − 1.00·8-s + (0.569 − 0.569i)9-s + (−1.24 − 0.514i)11-s + (0.169 + 0.408i)12-s + (0.196 − 0.0812i)13-s + (0.377 − 0.377i)14-s + 1.00·16-s − 0.685·17-s + (−0.569 + 0.569i)18-s + (−0.352 − 0.850i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.300372 - 0.393689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.300372 - 0.393689i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.292 - 0.707i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + (4.12 + 1.70i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.292i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (1.53 + 3.70i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (5.82 + 5.82i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.12 + 1.29i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.292i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (0.171 - 0.171i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.94 + 4.70i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + (0.464 - 1.12i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.87 + 4.53i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.70 + 0.707i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (2.29 + 5.53i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (5.82 + 5.82i)T + 71iT^{2} \) |
| 73 | \( 1 + (7 + 7i)T + 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-4.53 + 1.87i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.65 + 8.65i)T + 89iT^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−10.07225409748685699797420133923, −9.071834702998275022779416541806, −8.573723384227398633107604219688, −7.61710249737148169314875060431, −6.60955456609439166783867240541, −5.88745484842076707227250454662, −4.52692022777735404628327821111, −3.19268788643433014718892185210, −2.23903773794713734637222249752, −0.31559933256181672543311472512,
1.60413881787714813896679488978, 2.56330722072778295793989448181, 3.98257246566848786419816149631, 5.40326245755909615426051525736, 6.46940220170287833789781715992, 7.39209122095988472753278432523, 7.84453431084287399471193665931, 8.709496014269729955582586102010, 9.907575820298720015438336202501, 10.23202449528581467297999658553