| L(s) = 1 | + 0.510i·2-s + 1.73·4-s + 4.01·5-s + 1.90i·8-s + 2.04i·10-s − 4.55i·11-s − 1.36i·13-s + 2.50·16-s − 2.15·17-s − 2.66i·19-s + 6.99·20-s + 2.32·22-s + 1.27i·23-s + 11.1·25-s + 0.697·26-s + ⋯ |
| L(s) = 1 | + 0.360i·2-s + 0.869·4-s + 1.79·5-s + 0.674i·8-s + 0.647i·10-s − 1.37i·11-s − 0.379i·13-s + 0.626·16-s − 0.523·17-s − 0.610i·19-s + 1.56·20-s + 0.495·22-s + 0.266i·23-s + 2.22·25-s + 0.136·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.864370816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.864370816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.510iT - 2T^{2} \) |
| 5 | \( 1 - 4.01T + 5T^{2} \) |
| 11 | \( 1 + 4.55iT - 11T^{2} \) |
| 13 | \( 1 + 1.36iT - 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 2.66iT - 19T^{2} \) |
| 23 | \( 1 - 1.27iT - 23T^{2} \) |
| 29 | \( 1 + 8.75iT - 29T^{2} \) |
| 31 | \( 1 - 8.72iT - 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 - 7.89iT - 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 - 6.02iT - 61T^{2} \) |
| 67 | \( 1 - 8.15T + 67T^{2} \) |
| 71 | \( 1 - 0.301iT - 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 4.86T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 - 1.90T + 89T^{2} \) |
| 97 | \( 1 - 0.231iT - 97T^{2} \) |
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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
−9.722769368139770252001087632960, −8.777406268253667067092584371560, −8.117713764047990767263622416121, −6.88792889181714728867066804327, −6.28479545072084819559330966169, −5.73275613110715025165070951339, −4.94789870713395193368274543767, −3.17152368796468144291551769710, −2.43371177566107825982271520271, −1.32211811409782868576566555820,
1.70296342793192724233233070488, 2.01266101083696250753129380024, 3.15598420886883839715001679011, 4.59343054600593378343867549926, 5.50370585583802120529639133040, 6.52167984430618820353006461696, 6.77857864950742768150380629494, 7.940430994085696575283357607441, 9.154100340012850907187081904145, 9.846213216818646798309337103662
