L(s) = 1 | + (−0.866 + 0.5i)2-s + (1.67 − 0.448i)3-s + (0.499 − 0.866i)4-s + (−1.22 − 2.12i)5-s + (−1.22 + 1.22i)6-s + 0.999i·8-s + (2.59 − 1.50i)9-s + (2.12 + 1.22i)10-s + (0.448 − 1.67i)12-s − 2.44i·13-s + (−3 − 3i)15-s + (−0.5 − 0.866i)16-s + (2.44 − 4.24i)17-s + (−1.5 + 2.59i)18-s + (2.12 − 1.22i)19-s − 2.44·20-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.965 − 0.258i)3-s + (0.249 − 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.499 + 0.499i)6-s + 0.353i·8-s + (0.866 − 0.5i)9-s + (0.670 + 0.387i)10-s + (0.129 − 0.482i)12-s − 0.679i·13-s + (−0.774 − 0.774i)15-s + (−0.125 − 0.216i)16-s + (0.594 − 1.02i)17-s + (−0.353 + 0.612i)18-s + (0.486 − 0.280i)19-s − 0.547·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10514 - 0.499355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10514 - 0.499355i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.67 + 0.448i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 1.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-2.44 - 4.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.12 - 10.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.6 - 6.12i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−11.82810247261490600793329476880, −10.41938922768894230489577531533, −9.414378340257596038124887326459, −8.776918316112373973594954259257, −7.80760468819876549509842135260, −7.30666136809776217021982084888, −5.71272388282380583430050890182, −4.42195105338925144156065284308, −2.94385713946796282297852174789, −1.10146646405153012468474187965,
2.06414230689484616604091063502, 3.33439709888217118362448975755, 4.20511929541959119831960330478, 6.26610404456665463584090114507, 7.48255710469312579743049589896, 8.038614889912733326813571242535, 9.123536353662246039596239345982, 10.05529420907505693618927821442, 10.72565375125270546556252195292, 11.74346277787684775420462027651