L(s) = 1 | − i·3-s + (1.17 − 1.90i)5-s + 2.37i·7-s − 9-s − 2.77·11-s + 0.720i·13-s + (−1.90 − 1.17i)15-s − 3.36i·17-s − 5.08·19-s + 2.37·21-s + 6.99i·23-s + (−2.24 − 4.46i)25-s + i·27-s − 10.0·29-s + 7.55·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.524 − 0.851i)5-s + 0.898i·7-s − 0.333·9-s − 0.835·11-s + 0.199i·13-s + (−0.491 − 0.302i)15-s − 0.816i·17-s − 1.16·19-s + 0.518·21-s + 1.45i·23-s + (−0.449 − 0.893i)25-s + 0.192i·27-s − 1.87·29-s + 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.188881413\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.188881413\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.17 + 1.90i)T \) |
| 67 | \( 1 + iT \) |
good | 7 | \( 1 - 2.37iT - 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 0.720iT - 13T^{2} \) |
| 17 | \( 1 + 3.36iT - 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 - 6.99iT - 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 7.57iT - 43T^{2} \) |
| 47 | \( 1 + 0.513iT - 47T^{2} \) |
| 53 | \( 1 + 2.32iT - 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 4.90iT - 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 - 7.91iT - 83T^{2} \) |
| 89 | \( 1 + 3.49T + 89T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−8.404965709057482150870003303456, −8.041378415766855612973351593063, −7.11642044326788913780034931704, −6.22739983301776460282251053009, −5.56759379693784364500503992452, −5.07095033011746742162980078348, −4.08483179159166243787318531133, −2.74831370362284546782774261482, −2.17266300010734905636648835108, −1.11870476454522354138933165296,
0.34818539364610526347647978357, 2.06782887316312477579950745751, 2.73634364546899680309050353904, 3.90580908092707435826060994226, 4.27015441468156921403761839581, 5.50599585672865093376582104657, 6.00959732485339861355219246247, 6.90087464604906641311709408751, 7.52202822788701494425115969795, 8.345726236056787045959444797936