L(s) = 1 | + (1.58 + 1.14i)5-s + (−0.413 + 1.27i)7-s + (−0.809 + 0.587i)9-s + (−0.669 + 0.743i)11-s + (−1.47 − 1.07i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.873 + 2.68i)25-s + (−2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (0.0646 + 0.198i)47-s + (−0.639 − 0.464i)49-s + (−1.91 + 0.406i)55-s + (0.169 + 0.122i)61-s + ⋯ |
L(s) = 1 | + (1.58 + 1.14i)5-s + (−0.413 + 1.27i)7-s + (−0.809 + 0.587i)9-s + (−0.669 + 0.743i)11-s + (−1.47 − 1.07i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.873 + 2.68i)25-s + (−2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (0.0646 + 0.198i)47-s + (−0.639 − 0.464i)49-s + (−1.91 + 0.406i)55-s + (0.169 + 0.122i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.275969427\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.275969427\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.413 - 1.27i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.209T + T^{2} \) |
| 47 | \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−9.202243063460902253597869576461, −8.583148044008548032903930142228, −7.24419260628998080546133504629, −6.79284631312826339341559086613, −6.04673708764109857416908576283, −5.30365727230027360139275189329, −4.84264316476964723652985618673, −2.90712465344457956509136899414, −2.62715049272028536884273396896, −2.08463150631497355940375962182,
0.70597611889159737030225661894, 1.78964576931872483813959848984, 2.88967321233264231823417393549, 3.93305958075484151937457393960, 4.80656153291367570739118564268, 5.63816409435411985528918950259, 6.22364538283782206696572973138, 6.80933374820291450294857206446, 8.052182037163633557014946766231, 8.747674307957724746648128723993