| L(s) = 1 | − 3-s + 3.69·5-s + 7-s + 9-s − 3.21·11-s − 5.08·13-s − 3.69·15-s + 0.616·17-s − 4.48·19-s − 21-s + 1.38·23-s + 8.67·25-s − 27-s + 5.67·29-s − 6.91·31-s + 3.21·33-s + 3.69·35-s − 6.91·37-s + 5.08·39-s + 0.616·41-s + 7.99·43-s + 3.69·45-s − 4.97·47-s + 49-s − 0.616·51-s + 4.48·53-s − 11.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.65·5-s + 0.377·7-s + 0.333·9-s − 0.968·11-s − 1.40·13-s − 0.954·15-s + 0.149·17-s − 1.02·19-s − 0.218·21-s + 0.288·23-s + 1.73·25-s − 0.192·27-s + 1.05·29-s − 1.24·31-s + 0.559·33-s + 0.625·35-s − 1.13·37-s + 0.813·39-s + 0.0963·41-s + 1.21·43-s + 0.551·45-s − 0.725·47-s + 0.142·49-s − 0.0863·51-s + 0.616·53-s − 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 0.616T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 0.616T + 41T^{2} \) |
| 43 | \( 1 - 7.99T + 43T^{2} \) |
| 47 | \( 1 + 4.97T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−7.60810289915155453497682032324, −7.11134842448041200597174572727, −6.17189250762083520178077454912, −5.69299231241909078459653198276, −4.97346970304828038406769838608, −4.52488618996708150573079437720, −2.97322287032801366820795867171, −2.25897333164849208723750012690, −1.51755378429024699932420008006, 0,
1.51755378429024699932420008006, 2.25897333164849208723750012690, 2.97322287032801366820795867171, 4.52488618996708150573079437720, 4.97346970304828038406769838608, 5.69299231241909078459653198276, 6.17189250762083520178077454912, 7.11134842448041200597174572727, 7.60810289915155453497682032324
