| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4.52·7-s + 8-s + 9-s − 10-s + 0.738·11-s − 12-s + 2.22·13-s − 4.52·14-s + 15-s + 16-s + 18-s − 3.49·19-s − 20-s + 4.52·21-s + 0.738·22-s + 4.95·23-s − 24-s + 25-s + 2.22·26-s − 27-s − 4.52·28-s + 2.39·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.70·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.222·11-s − 0.288·12-s + 0.616·13-s − 1.20·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s − 0.801·19-s − 0.223·20-s + 0.986·21-s + 0.157·22-s + 1.03·23-s − 0.204·24-s + 0.200·25-s + 0.436·26-s − 0.192·27-s − 0.854·28-s + 0.445·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 - 0.738T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 + 9.10T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 0.452T + 47T^{2} \) |
| 53 | \( 1 - 0.507T + 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 + 6.20T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 + 2.39T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 5.22T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 8.53T + 97T^{2} \) |
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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
−7.10290472833651186374898671833, −6.62306703557894179626367883126, −6.07834187345298707794995807605, −5.47900686167791712464816457622, −4.51320652763996233943000273075, −3.87171025309323936992600127856, −3.26492281530651128594794605840, −2.48169981115629039900512089962, −1.13401038559153785182748544402, 0,
1.13401038559153785182748544402, 2.48169981115629039900512089962, 3.26492281530651128594794605840, 3.87171025309323936992600127856, 4.51320652763996233943000273075, 5.47900686167791712464816457622, 6.07834187345298707794995807605, 6.62306703557894179626367883126, 7.10290472833651186374898671833
