L(s) = 1 | − i·2-s + (0.271 + 1.71i)3-s − 4-s + (0.5 − 0.866i)5-s + (1.71 − 0.271i)6-s + (1.13 + 2.38i)7-s + i·8-s + (−2.85 + 0.929i)9-s + (−0.866 − 0.5i)10-s + (−4.25 + 2.45i)11-s + (−0.271 − 1.71i)12-s + (−3.91 + 2.26i)13-s + (2.38 − 1.13i)14-s + (1.61 + 0.619i)15-s + 16-s + (−2.06 + 3.57i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.156 + 0.987i)3-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.698 − 0.110i)6-s + (0.429 + 0.903i)7-s + 0.353i·8-s + (−0.950 + 0.309i)9-s + (−0.273 − 0.158i)10-s + (−1.28 + 0.740i)11-s + (−0.0784 − 0.493i)12-s + (−1.08 + 0.627i)13-s + (0.638 − 0.303i)14-s + (0.417 + 0.160i)15-s + 0.250·16-s + (−0.501 + 0.868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641393 + 0.774277i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641393 + 0.774277i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.271 - 1.71i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.13 - 2.38i)T \) |
good | 11 | \( 1 + (4.25 - 2.45i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.91 - 2.26i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.06 - 3.57i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.89 + 3.98i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.36 + 2.51i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.46 + 1.42i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.57iT - 31T^{2} \) |
| 37 | \( 1 + (-5.05 - 8.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.31 - 7.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.88 - 4.99i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 0.454T + 47T^{2} \) |
| 53 | \( 1 + (-10.7 - 6.21i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 4.23iT - 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 + 8.14iT - 71T^{2} \) |
| 73 | \( 1 + (-2.06 - 1.19i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 + (-1.42 + 2.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.95 + 5.11i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.74 + 1.00i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.78607239138212680262505963630, −9.761990838961973390311631336829, −9.507639170021935652798314853159, −8.452796822659743487521341308836, −7.68379279477222676335334399828, −5.94911167726665792809683787961, −4.85972307438568184667037512315, −4.61896895197625281054484237653, −2.91648722234640313429153909578, −2.13099858038019298583216885548,
0.51169833479378669992323029045, 2.36957846056541195858136734158, 3.58809138457885890662863098873, 5.27608238836359619373785942217, 5.74869095236079690027554463072, 7.20392635836160302744577768092, 7.50447691995977248661156549884, 8.136852476735843047593727485477, 9.410063532490205019547855502361, 10.29438888307435901887103293617