L(s) = 1 | − 1.41·2-s + (−1.70 + 0.707i)3-s + 2.00·4-s + (2.41 − 1.00i)6-s + (1 − i)7-s − 2.82·8-s + (0.292 − 0.292i)9-s + (0.121 − 0.292i)11-s + (−3.41 + 1.41i)12-s + (0.707 + 1.70i)13-s + (−1.41 + 1.41i)14-s + 4.00·16-s − 2.82·17-s + (−0.414 + 0.414i)18-s + (5.53 − 2.29i)19-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (−0.985 + 0.408i)3-s + 1.00·4-s + (0.985 − 0.408i)6-s + (0.377 − 0.377i)7-s − 1.00·8-s + (0.0976 − 0.0976i)9-s + (0.0365 − 0.0883i)11-s + (−0.985 + 0.408i)12-s + (0.196 + 0.473i)13-s + (−0.377 + 0.377i)14-s + 1.00·16-s − 0.685·17-s + (−0.0976 + 0.0976i)18-s + (1.26 − 0.526i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.624797 + 0.214007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624797 + 0.214007i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.70 - 0.707i)T + (2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.121 + 0.292i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 1.70i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-5.53 + 2.29i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.171 - 0.171i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.12 + 2.70i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.70i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.82 - 5.82i)T - 41iT^{2} \) |
| 43 | \( 1 + (-7.94 - 3.29i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + (-7.53 - 3.12i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-6.12 - 2.53i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.292 - 0.707i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-3.70 + 1.53i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (0.171 + 0.171i)T + 71iT^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-2.53 - 6.12i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.65 - 2.65i)T + 89iT^{2} \) |
| 97 | \( 1 - 1.51iT - 97T^{2} \) |
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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
−10.44007133376900867379545703259, −9.560323940087322706129128458803, −8.823378299432546918540945900126, −7.76849004393184041847322894892, −6.99940320574306605003323682531, −6.03740067475674015265280058559, −5.20377785973001126816566215107, −4.02918467664771298869575540197, −2.48114800120637034124241801980, −0.913695392445522373512712512376,
0.72503249664730898008217324124, 2.05149239993406909570954934498, 3.48837748194051890163937348067, 5.29786222583280730783864771199, 5.81871975790082292275093956114, 6.87427555025788493452017104273, 7.51033371827293704919441717895, 8.583357245971106911410697956766, 9.229082753424442611042463472877, 10.33838458820306011441008137460