L(s) = 1 | + (−0.0647 − 1.99i)2-s + (−3.11 + 1.79i)3-s + (−3.99 + 0.259i)4-s + (4.52 + 7.84i)5-s + (3.79 + 6.11i)6-s − 2.81i·7-s + (0.776 + 7.96i)8-s + (1.97 − 3.41i)9-s + (15.3 − 9.56i)10-s + 11.6i·11-s + (11.9 − 7.98i)12-s + (−2.20 + 3.81i)13-s + (−5.63 + 0.182i)14-s + (−28.2 − 16.2i)15-s + (15.8 − 2.06i)16-s + (−9.59 − 16.6i)17-s + ⋯ |
L(s) = 1 | + (−0.0323 − 0.999i)2-s + (−1.03 + 0.599i)3-s + (−0.997 + 0.0647i)4-s + (0.905 + 1.56i)5-s + (0.632 + 1.01i)6-s − 0.402i·7-s + (0.0970 + 0.995i)8-s + (0.218 − 0.379i)9-s + (1.53 − 0.956i)10-s + 1.06i·11-s + (0.997 − 0.665i)12-s + (−0.169 + 0.293i)13-s + (−0.402 + 0.0130i)14-s + (−1.88 − 1.08i)15-s + (0.991 − 0.129i)16-s + (−0.564 − 0.977i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.723842 + 0.360929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.723842 + 0.360929i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0647 + 1.99i)T \) |
| 19 | \( 1 + (9.06 - 16.6i)T \) |
good | 3 | \( 1 + (3.11 - 1.79i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-4.52 - 7.84i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + 2.81iT - 49T^{2} \) |
| 11 | \( 1 - 11.6iT - 121T^{2} \) |
| 13 | \( 1 + (2.20 - 3.81i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (9.59 + 16.6i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-25.2 - 14.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-15.9 + 27.5i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 23.9iT - 961T^{2} \) |
| 37 | \( 1 - 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-2.87 - 4.97i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-12.4 + 7.18i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-36.4 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-16.2 + 28.2i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (9.13 - 5.27i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (53.5 - 92.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-109. - 63.4i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-3.89 + 2.24i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (12.5 + 21.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (70.4 - 40.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 138. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (24.1 - 41.8i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (26.7 + 46.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−14.24826460924156173018577568046, −13.31399307927629812064592118607, −11.79420547571958899006075182146, −10.99138014282990332444198097113, −10.20616840253269617719936894448, −9.566333539788633353251627108531, −7.21287905594457733941492668056, −5.79287866959648804700539142472, −4.35238116688411334637843118217, −2.45820122416354919695798815238,
0.833514306498654404671738257528, 4.87577378217656175505112878902, 5.72376243296096437577119124098, 6.60310650914980973545269121463, 8.510935537632425714693119871631, 9.070807238367828124866368372446, 10.78047047237600098012694221049, 12.51000300936803134741848623790, 12.86817798192585027925035622654, 13.92423970247059737533092741689