L(s) = 1 | + (−0.577 + 1.29i)2-s + (−1.33 − 1.49i)4-s + (3.11 + 1.80i)5-s + (−0.838 − 2.50i)7-s + (2.69 − 0.861i)8-s + (−4.12 + 2.98i)10-s + (−4.13 + 2.38i)11-s + 2.81i·13-s + (3.72 + 0.365i)14-s + (−0.443 + 3.97i)16-s + (1.10 − 0.640i)17-s + (−1.39 + 2.42i)19-s + (−1.47 − 7.04i)20-s + (−0.694 − 6.70i)22-s + (5.58 + 3.22i)23-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.912i)2-s + (−0.666 − 0.745i)4-s + (1.39 + 0.805i)5-s + (−0.317 − 0.948i)7-s + (0.952 − 0.304i)8-s + (−1.30 + 0.944i)10-s + (−1.24 + 0.719i)11-s + 0.781i·13-s + (0.995 + 0.0976i)14-s + (−0.110 + 0.993i)16-s + (0.269 − 0.155i)17-s + (−0.320 + 0.555i)19-s + (−0.329 − 1.57i)20-s + (−0.148 − 1.43i)22-s + (1.16 + 0.672i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.714208 + 1.07989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.714208 + 1.07989i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.577 - 1.29i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.838 + 2.50i)T \) |
good | 5 | \( 1 + (-3.11 - 1.80i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.13 - 2.38i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.81iT - 13T^{2} \) |
| 17 | \( 1 + (-1.10 + 0.640i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 2.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.58 - 3.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 + (-5.30 - 9.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.554 - 0.960i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.06iT - 41T^{2} \) |
| 43 | \( 1 + 9.45iT - 43T^{2} \) |
| 47 | \( 1 + (2.18 - 3.77i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.06 - 5.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.957 + 1.65i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.59 + 4.96i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.12 + 4.68i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.12iT - 71T^{2} \) |
| 73 | \( 1 + (7.42 - 4.28i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.33 + 1.34i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.62T + 83T^{2} \) |
| 89 | \( 1 + (-4.78 - 2.76i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.51iT - 97T^{2} \) |
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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
−10.26871168895536189597549579419, −9.918613864011921007306978730562, −8.986573356415049254319457958022, −7.83635291904592009713959658516, −6.91726391576680992707579815577, −6.55307731060219395396670498409, −5.42342580009705847072060560638, −4.59808439877346461261340265853, −2.95735242122961623913283935787, −1.51881999334024386122194495314,
0.805674954966373351854462510396, 2.39699120653366520495211216377, 2.91054840780297100322885143478, 4.76577274633307109164469925081, 5.42580668962197005420600352739, 6.32116127580549727890903291363, 7.995962890816865579373438465506, 8.606447556253204468892834925375, 9.297476599266500098162886185834, 10.10682439238796382009574929102