| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s − 2·10-s + 4·11-s + 2·12-s − 13-s + 2·14-s − 15-s − 4·16-s − 5·17-s + 2·18-s − 6·19-s − 2·20-s + 21-s + 8·22-s + 25-s − 2·26-s + 27-s + 2·28-s + 9·29-s − 2·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s − 16-s − 1.21·17-s + 0.471·18-s − 1.37·19-s − 0.447·20-s + 0.218·21-s + 1.70·22-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.377·28-s + 1.67·29-s − 0.365·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.824130676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.824130676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.48271843457153, −13.91843249127766, −13.39453039552231, −13.04292520329490, −12.42277109285067, −11.97519886276307, −11.54622567859148, −11.10172050472719, −10.37511251330776, −9.835460224127161, −8.962351413983665, −8.703647014281345, −8.269948195753581, −7.399775808657748, −6.819741264537615, −6.359012124834661, −6.032852246908801, −4.855681233256496, −4.629985205767082, −4.209189032845916, −3.619332848259139, −2.949748807610243, −2.332412030378960, −1.717892286586642, −0.6121549816119179,
0.6121549816119179, 1.717892286586642, 2.332412030378960, 2.949748807610243, 3.619332848259139, 4.209189032845916, 4.629985205767082, 4.855681233256496, 6.032852246908801, 6.359012124834661, 6.819741264537615, 7.399775808657748, 8.269948195753581, 8.703647014281345, 8.962351413983665, 9.835460224127161, 10.37511251330776, 11.10172050472719, 11.54622567859148, 11.97519886276307, 12.42277109285067, 13.04292520329490, 13.39453039552231, 13.91843249127766, 14.48271843457153
