L(s) = 1 | + (1.58 + 0.916i)2-s + (0.678 + 1.17i)4-s + 0.645·5-s − 1.17i·8-s + (1.02 + 0.591i)10-s − 5.31i·11-s + (4.44 + 2.56i)13-s + (2.43 − 4.22i)16-s + (0.814 − 1.41i)17-s + (−2.09 + 1.20i)19-s + (0.437 + 0.758i)20-s + (4.86 − 8.43i)22-s − 1.47i·23-s − 4.58·25-s + (4.69 + 8.13i)26-s + ⋯ |
L(s) = 1 | + (1.12 + 0.647i)2-s + (0.339 + 0.587i)4-s + 0.288·5-s − 0.416i·8-s + (0.323 + 0.187i)10-s − 1.60i·11-s + (1.23 + 0.711i)13-s + (0.609 − 1.05i)16-s + (0.197 − 0.342i)17-s + (−0.479 + 0.276i)19-s + (0.0978 + 0.169i)20-s + (1.03 − 1.79i)22-s − 0.306i·23-s − 0.916·25-s + (0.921 + 1.59i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.242671552\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.242671552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.58 - 0.916i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 0.645T + 5T^{2} \) |
| 11 | \( 1 + 5.31iT - 11T^{2} \) |
| 13 | \( 1 + (-4.44 - 2.56i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.814 + 1.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.09 - 1.20i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.47iT - 23T^{2} \) |
| 29 | \( 1 + (-6.43 + 3.71i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.90 + 2.83i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.99 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.99 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.51 + 2.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.54 + 2.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.04 - 1.18i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.47 + 2.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.18 - 5.30i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.07 - 8.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (10.2 + 5.90i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.51 + 6.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.16 - 3.74i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.3 - 8.31i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.677778968651690390568244713724, −8.547365385427404760545219073970, −8.068881439430850105781856787330, −6.66090323057776082705595276361, −6.23926743221607537083192269949, −5.63386250461604189343571526802, −4.53937697941336542399265502143, −3.77803667968324387472616539459, −2.83237618436798500955229015959, −1.05422502546810697562361914530,
1.56466146185348498099094207714, 2.53623749079187314408482659166, 3.62381928273763714114122948711, 4.37765652025903798723431538448, 5.24821629447833155610497679826, 6.02998722858490267920833069340, 6.98882124364322127917222087735, 8.069890559206477078588048138249, 8.792680091842854163481949701568, 9.956783075738088146875178712744