| L(s) = 1 | − 2.63·3-s − 2.63·5-s − 3.88·7-s + 3.95·9-s − 5.95·11-s − 4.14·13-s + 6.94·15-s + 1.82·17-s + 19-s + 10.2·21-s + 4.14·23-s + 1.93·25-s − 2.50·27-s + 8.19·29-s − 1.67·31-s + 15.6·33-s + 10.2·35-s + 8.14·37-s + 10.9·39-s − 0.514·41-s − 10.9·43-s − 10.4·45-s + 13.3·47-s + 8.10·49-s − 4.82·51-s − 1.62·53-s + 15.6·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s − 1.17·5-s − 1.46·7-s + 1.31·9-s − 1.79·11-s − 1.15·13-s + 1.79·15-s + 0.443·17-s + 0.229·19-s + 2.23·21-s + 0.865·23-s + 0.387·25-s − 0.482·27-s + 1.52·29-s − 0.301·31-s + 2.73·33-s + 1.73·35-s + 1.33·37-s + 1.75·39-s − 0.0803·41-s − 1.67·43-s − 1.55·45-s + 1.94·47-s + 1.15·49-s − 0.675·51-s − 0.223·53-s + 2.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 + 0.514T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 2.42T + 79T^{2} \) |
| 83 | \( 1 + 0.987T + 83T^{2} \) |
| 89 | \( 1 + 2.25T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−7.55240229264295542769561252930, −7.25793270794560137145800285016, −6.41489718737997734532758340783, −5.70942290309791135236864852162, −4.97931162584221276279636106214, −4.44935839959076449448903409867, −3.24385568014908629284142880980, −2.68008441742664101693844973601, −0.68442339082680007335777975565, 0,
0.68442339082680007335777975565, 2.68008441742664101693844973601, 3.24385568014908629284142880980, 4.44935839959076449448903409867, 4.97931162584221276279636106214, 5.70942290309791135236864852162, 6.41489718737997734532758340783, 7.25793270794560137145800285016, 7.55240229264295542769561252930
