L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + 2i·5-s + (−0.866 − 0.499i)6-s + (3.46 + 2i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)10-s + (3.46 − 2i)11-s + 0.999·12-s − 3.99·14-s + (−1.73 + i)15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + 0.894i·5-s + (−0.353 − 0.204i)6-s + (1.30 + 0.755i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (1.04 − 0.603i)11-s + 0.288·12-s − 1.06·14-s + (−0.447 + 0.258i)15-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.674657195\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674657195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.46 + 2i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.92 - 4i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.66 - 5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.8 - 8i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.92 - 4i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (12.1 - 7i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.66 + 5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.936193974606221197965670422348, −9.387423561374392878630978863963, −8.436068708170155232079764512734, −7.87459788872227501240420878374, −6.95557565334194861427303394131, −5.87340093876526780398250693671, −5.18839788149181929584059259586, −3.86582321560939600417178721843, −2.76477429205027418481081924013, −1.49784687045497768096058282702,
1.14810231307952175467939165403, 1.60290371862982725135754429192, 3.27624232899731830590955182173, 4.45597645166152038457802064950, 5.19794544272011510995124710335, 6.72617146178320099045695754933, 7.45597155505084620545421606239, 8.105933640065323688362981131644, 8.988310900441996454595398524526, 9.464888196908609242086131943348