L(s) = 1 | + (−0.473 − 0.819i)2-s + (0.552 − 0.956i)4-s − 5-s + (0.781 − 1.35i)7-s − 2.93·8-s + (0.473 + 0.819i)10-s + (−0.754 − 1.30i)11-s + (−3.46 − 1.00i)13-s − 1.47·14-s + (0.285 + 0.494i)16-s + (−1.63 + 2.82i)17-s + (1.55 − 2.68i)19-s + (−0.552 + 0.956i)20-s + (−0.714 + 1.23i)22-s + (−1.46 − 2.54i)23-s + ⋯ |
L(s) = 1 | + (−0.334 − 0.579i)2-s + (0.276 − 0.478i)4-s − 0.447·5-s + (0.295 − 0.511i)7-s − 1.03·8-s + (0.149 + 0.259i)10-s + (−0.227 − 0.394i)11-s + (−0.960 − 0.278i)13-s − 0.395·14-s + (0.0714 + 0.123i)16-s + (−0.395 + 0.685i)17-s + (0.356 − 0.616i)19-s + (−0.123 + 0.213i)20-s + (−0.152 + 0.263i)22-s + (−0.306 − 0.530i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00487919 + 0.714612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00487919 + 0.714612i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (3.46 + 1.00i)T \) |
good | 2 | \( 1 + (0.473 + 0.819i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.781 + 1.35i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.754 + 1.30i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.63 - 2.82i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.55 + 2.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.46 + 2.54i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.30 + 2.25i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 + (0.285 + 0.494i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.84 + 4.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.632 - 1.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 + (-6.40 + 11.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 - 5.76i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.74 + 8.22i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.610 + 1.05i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + (-0.398 - 0.690i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.90 + 5.02i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.44006969736153160824163506075, −9.571085743284041791780027471078, −8.622416044080007475510203032872, −7.64334972849307288634220556423, −6.72651609181756424780771332990, −5.62498182177589126322734301153, −4.55041612934241816305549533797, −3.23577558499124725871142205990, −2.02439802509190014260273124543, −0.41813823383855153823387318029,
2.21375426964418622656186514437, 3.39612450331145986062101219649, 4.73575277624508355585083380408, 5.76505204334947898041086879130, 7.00079331348533549743022903656, 7.50133362205321470285851815253, 8.381729491761145029826440970228, 9.197018374774659661626995747696, 10.07992692568244791215167353910, 11.39914533504032866057360545520