L(s) = 1 | + (0.590 + 2.20i)3-s + (1.73 + 1.40i)5-s + (−2.34 + 1.22i)7-s + (−1.90 + 1.10i)9-s + (−0.0644 + 0.111i)11-s + (0.748 − 0.748i)13-s + (−2.07 + 4.65i)15-s + (2.28 − 0.613i)17-s + (−3.81 − 6.59i)19-s + (−4.08 − 4.44i)21-s + (−1.10 + 4.11i)23-s + (1.02 + 4.89i)25-s + (1.28 + 1.28i)27-s + 0.163i·29-s + (8.43 + 4.86i)31-s + ⋯ |
L(s) = 1 | + (0.340 + 1.27i)3-s + (0.776 + 0.630i)5-s + (−0.886 + 0.462i)7-s + (−0.636 + 0.367i)9-s + (−0.0194 + 0.0336i)11-s + (0.207 − 0.207i)13-s + (−0.536 + 1.20i)15-s + (0.555 − 0.148i)17-s + (−0.874 − 1.51i)19-s + (−0.890 − 0.970i)21-s + (−0.229 + 0.857i)23-s + (0.205 + 0.978i)25-s + (0.246 + 0.246i)27-s + 0.0303i·29-s + (1.51 + 0.874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.945581 + 1.10814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.945581 + 1.10814i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 - 1.40i)T \) |
| 7 | \( 1 + (2.34 - 1.22i)T \) |
good | 3 | \( 1 + (-0.590 - 2.20i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (0.0644 - 0.111i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.748 + 0.748i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.28 + 0.613i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.81 + 6.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.10 - 4.11i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 0.163iT - 29T^{2} \) |
| 31 | \( 1 + (-8.43 - 4.86i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0900 + 0.0241i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 11.9iT - 41T^{2} \) |
| 43 | \( 1 + (-2.76 - 2.76i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.597 + 2.23i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.32 + 1.96i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.13 + 7.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (11.5 - 6.64i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.83 + 6.84i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 + (-0.483 - 1.80i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-8.48 + 4.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.43 + 8.43i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.37 + 5.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.61 - 9.61i)T + 97iT^{2} \) |
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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
−12.04405041869427657510124079483, −10.75930957450259788742666727365, −10.20265487451954484828899253731, −9.373967095554122489546483248168, −8.759193781704226268553021145963, −7.07873948827184878451932285255, −6.05090168995123394718848742410, −4.95981325649096286352816728704, −3.55616168790708888411975977672, −2.62338728966307810270044280699,
1.19712959400371256527270255638, 2.55708205476017672909195899664, 4.24047241533126765413798354246, 6.01471530287465809875265993637, 6.48905692346731526632943044566, 7.78714841594319234013704224178, 8.522047868165479807723125690354, 9.738775336002595105094360291355, 10.44040716435896192530039282357, 12.11263284298570929456520392452