| L(s) = 1 | − 6.25·3-s + 5·5-s + 34.6·7-s + 12.1·9-s + 7.91·11-s − 11.0·13-s − 31.2·15-s + 57.6·17-s − 141.·19-s − 216.·21-s − 129.·23-s + 25·25-s + 92.8·27-s − 89.1·29-s − 78.3·31-s − 49.5·33-s + 173.·35-s − 249.·37-s + 69.0·39-s − 91.8·41-s − 194.·43-s + 60.8·45-s − 72.6·47-s + 856.·49-s − 360.·51-s + 456.·53-s + 39.5·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s + 0.447·5-s + 1.86·7-s + 0.450·9-s + 0.216·11-s − 0.235·13-s − 0.538·15-s + 0.821·17-s − 1.70·19-s − 2.25·21-s − 1.17·23-s + 0.200·25-s + 0.661·27-s − 0.570·29-s − 0.453·31-s − 0.261·33-s + 0.836·35-s − 1.10·37-s + 0.283·39-s − 0.349·41-s − 0.688·43-s + 0.201·45-s − 0.225·47-s + 2.49·49-s − 0.989·51-s + 1.18·53-s + 0.0970·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 3 | \( 1 + 6.25T + 27T^{2} \) |
| 7 | \( 1 - 34.6T + 343T^{2} \) |
| 11 | \( 1 - 7.91T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 249.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 72.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 341.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 217.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 529.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 381.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 876.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 203.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 996.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 172.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.715890282301864804592449241391, −8.146538134892244175728865020082, −7.16557271361858943288172744589, −6.19415002912382710027699295126, −5.45884919390614249245428273832, −4.87065533214288845159890727520, −3.96048564418966408050180800300, −2.16627010265726148798276350626, −1.38459410927332807967246938330, 0,
1.38459410927332807967246938330, 2.16627010265726148798276350626, 3.96048564418966408050180800300, 4.87065533214288845159890727520, 5.45884919390614249245428273832, 6.19415002912382710027699295126, 7.16557271361858943288172744589, 8.146538134892244175728865020082, 8.715890282301864804592449241391
