L(s) = 1 | + (−0.411 + 1.35i)2-s + (−0.755 + 0.654i)3-s + (−1.66 − 1.11i)4-s + (−2.07 + 0.298i)5-s + (−0.575 − 1.29i)6-s + (1.09 − 2.39i)7-s + (2.19 − 1.78i)8-s + (0.142 − 0.989i)9-s + (0.450 − 2.93i)10-s + (−0.479 + 1.63i)11-s + (1.98 − 0.246i)12-s + (2.58 − 1.17i)13-s + (2.79 + 2.47i)14-s + (1.37 − 1.58i)15-s + (1.52 + 3.69i)16-s + (2.37 + 1.52i)17-s + ⋯ |
L(s) = 1 | + (−0.290 + 0.956i)2-s + (−0.436 + 0.378i)3-s + (−0.830 − 0.556i)4-s + (−0.928 + 0.133i)5-s + (−0.234 − 0.527i)6-s + (0.414 − 0.907i)7-s + (0.774 − 0.632i)8-s + (0.0474 − 0.329i)9-s + (0.142 − 0.926i)10-s + (−0.144 + 0.492i)11-s + (0.572 − 0.0711i)12-s + (0.716 − 0.327i)13-s + (0.747 + 0.660i)14-s + (0.354 − 0.409i)15-s + (0.380 + 0.924i)16-s + (0.574 + 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.732100 + 0.478659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.732100 + 0.478659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.411 - 1.35i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-4.78 - 0.359i)T \) |
good | 5 | \( 1 + (2.07 - 0.298i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 2.39i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.479 - 1.63i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.58 + 1.17i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 1.52i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.187 - 0.291i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.23 - 3.47i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.65 + 4.22i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-7.11 - 1.02i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.262 - 1.82i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.53 + 2.19i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + (-2.30 - 1.05i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-5.62 + 2.56i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.14 - 4.45i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.83 + 6.25i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-14.6 + 4.30i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.51 - 3.54i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.00 + 10.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-3.63 - 0.522i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.14 - 7.08i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.676 - 4.70i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.79499133766984417169145948756, −10.11021441636484616868390387203, −9.084765742977411752164984409482, −7.963311537602999757288543729574, −7.51747204333258982428664497715, −6.51735889102691756096501401203, −5.42050508543830894487679986022, −4.41341370143026245949991057174, −3.67258675390546411689964016491, −0.927493110714430225065790279013,
0.925814514167204739142428731263, 2.49371695112540636006859820200, 3.72186889202890900888893372377, 4.84802430443745057126352998840, 5.83709822215954887106597265912, 7.29018961176559817152633275954, 8.222211581537046558086518270388, 8.751848902895044150430914471175, 9.825587315698004152982141462806, 11.03965536998969523374018913677