L(s) = 1 | + (2.59 + 1.5i)3-s + (−1.86 − 1.23i)5-s + (3 + 5.19i)9-s + (−1.5 + 2.59i)11-s − i·13-s + (−3 − 6i)15-s + (4.33 + 2.5i)17-s + (4 + 6.92i)19-s + (1.73 − i)23-s + (1.96 + 4.59i)25-s + 9i·27-s + 29-s + (−1 + 1.73i)31-s + (−7.79 + 4.5i)33-s + (−8.66 + 5i)37-s + ⋯ |
L(s) = 1 | + (1.49 + 0.866i)3-s + (−0.834 − 0.550i)5-s + (1 + 1.73i)9-s + (−0.452 + 0.783i)11-s − 0.277i·13-s + (−0.774 − 1.54i)15-s + (1.05 + 0.606i)17-s + (0.917 + 1.58i)19-s + (0.361 − 0.208i)23-s + (0.392 + 0.919i)25-s + 1.73i·27-s + 0.185·29-s + (−0.179 + 0.311i)31-s + (−1.35 + 0.783i)33-s + (−1.42 + 0.821i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84938 + 1.34465i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84938 + 1.34465i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (-4.33 - 2.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.66 - 5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (-9.52 + 5.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (8.66 + 5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−10.13733627562758213459076830870, −9.255980668023784122185753333510, −8.473582595800703477796539695674, −7.86823385909909707634155860447, −7.31334669445838623682639990113, −5.52920260136622659280937084706, −4.66841974426817634879557763709, −3.71608156313355205832902244823, −3.16506432793313423237778978685, −1.68824620037295453690926620688,
0.966976187334345446841872998318, 2.72016682766894819974051636666, 3.04920154635995798618464664738, 4.15573363442531655335855105067, 5.59450076491160364774826823158, 7.00048018237222261760213757137, 7.33424384162469435487554391358, 8.058628622037710036541747394354, 8.883375524921561574878581081496, 9.498201409521387193095403759754