| L(s) = 1 | − 1.80i·2-s − 0.561i·3-s − 1.25·4-s + (−0.178 − 2.22i)5-s − 1.01·6-s + 3.11i·7-s − 1.34i·8-s + 2.68·9-s + (−4.01 + 0.321i)10-s − 5.61·11-s + 0.703i·12-s + 1.24i·13-s + 5.61·14-s + (−1.25 + 0.100i)15-s − 4.93·16-s + ⋯ |
| L(s) = 1 | − 1.27i·2-s − 0.324i·3-s − 0.626·4-s + (−0.0796 − 0.996i)5-s − 0.413·6-s + 1.17i·7-s − 0.476i·8-s + 0.894·9-s + (−1.27 + 0.101i)10-s − 1.69·11-s + 0.203i·12-s + 0.346i·13-s + 1.49·14-s + (−0.323 + 0.0258i)15-s − 1.23·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5281995747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5281995747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.178 + 2.22i)T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.80iT - 2T^{2} \) |
| 3 | \( 1 + 0.561iT - 3T^{2} \) |
| 7 | \( 1 - 3.11iT - 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 - 1.24iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2.32iT - 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 - 1.90iT - 37T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 - 2.04iT - 43T^{2} \) |
| 47 | \( 1 + 4.85iT - 47T^{2} \) |
| 53 | \( 1 + 9.11iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 8.46iT - 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 1.56iT - 73T^{2} \) |
| 79 | \( 1 - 4.91T + 79T^{2} \) |
| 83 | \( 1 - 3.94iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−9.107423508847395826202542639438, −8.349077703114772634896844755732, −7.51509459494567180323475018110, −6.40102770068463814110379978601, −5.31506757686974333587922650625, −4.62450926391204792615116985542, −3.53701577490347268637141418718, −2.26278586586650891631245636533, −1.81163493190699557251385266472, −0.19059573793131521051834249299,
2.13380630298043453059573986944, 3.42702900355250888737398816116, 4.39174751880946961855759722451, 5.30969205085939295447547132646, 6.16924730688407643678359690870, 7.14496395030185312949922233893, 7.48101120322767627993840408366, 8.011886429356934704579080719973, 9.256694185231227719910744606296, 10.29127679603532216063845363353
