| L(s) = 1 | + 2.73·3-s − 0.732i·7-s + 4.46·9-s − 2i·11-s + 3.46·13-s + 3.46i·17-s − 0.535i·19-s − 2i·21-s − 6.19i·23-s + 3.99·27-s + 6.92i·29-s + 5.46·31-s − 5.46i·33-s − 2·37-s + 9.46·39-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.276i·7-s + 1.48·9-s − 0.603i·11-s + 0.960·13-s + 0.840i·17-s − 0.122i·19-s − 0.436i·21-s − 1.29i·23-s + 0.769·27-s + 1.28i·29-s + 0.981·31-s − 0.951i·33-s − 0.328·37-s + 1.51·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.68187 - 0.271588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.68187 - 0.271588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 0.535iT - 19T^{2} \) |
| 23 | \( 1 + 6.19iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 3.26iT - 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.46iT - 59T^{2} \) |
| 61 | \( 1 - 8.92iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46iT - 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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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
−10.26449756023322572318091105717, −9.028106735328185235953763171036, −8.615077055693976445703497036036, −7.968502777886946179816400226733, −6.93117411464043342299825951590, −5.98749488844225468402154746379, −4.49762658742986309713201419060, −3.58497330970638344912059166494, −2.78690253514986451073744104943, −1.44042933670270475032837919096,
1.64377486087701292361627624841, 2.75731264472808744127891687876, 3.63341304767975850645846690948, 4.65616410189221536010085288205, 5.97053639946335435765034744529, 7.11557771179290550011259327190, 7.917567531033544188329123125249, 8.545422724768077970584142903971, 9.486656344567439707806042267943, 9.829395730487599738624468114762
