L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 − 1.22i)13-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (0.965 − 0.258i)23-s + (1.49 + 0.866i)26-s + (−0.258 + 0.965i)28-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 − 1.22i)13-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (0.965 − 0.258i)23-s + (1.49 + 0.866i)26-s + (−0.258 + 0.965i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.279782272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.279782272\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.907996749547684052946125264276, −8.642037047293212590643431079003, −8.053152222904110516732244928517, −7.66442818014527895930106300616, −6.54323637870039922732932775900, −5.78671244300403476087012358938, −4.93800235149329342689794437392, −4.06130047328817027678320173259, −3.20376541789548576100150409441, −1.25999334972318086574075162043,
1.49096835207782934262291208101, 2.29848189773770337800704330594, 3.59638645724153467551674027748, 4.57267636824737300026341415992, 5.05263577622231151367012144439, 6.33996708144374766729944158837, 7.08930800466539906092413884479, 8.440138412707018885313531893278, 8.971814698362603903124167839806, 9.640521003436648658714820385392