| L(s) = 1 | − 2-s + 2.82·3-s + 4-s − 1.73·5-s − 2.82·6-s + 7-s − 8-s + 4.97·9-s + 1.73·10-s − 2.33·11-s + 2.82·12-s + 0.236·13-s − 14-s − 4.89·15-s + 16-s − 4.82·17-s − 4.97·18-s + 3.78·19-s − 1.73·20-s + 2.82·21-s + 2.33·22-s − 0.553·23-s − 2.82·24-s − 1.99·25-s − 0.236·26-s + 5.58·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.775·5-s − 1.15·6-s + 0.377·7-s − 0.353·8-s + 1.65·9-s + 0.548·10-s − 0.704·11-s + 0.815·12-s + 0.0655·13-s − 0.267·14-s − 1.26·15-s + 0.250·16-s − 1.17·17-s − 1.17·18-s + 0.868·19-s − 0.387·20-s + 0.616·21-s + 0.497·22-s − 0.115·23-s − 0.576·24-s − 0.398·25-s − 0.0463·26-s + 1.07·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 0.553T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 1.56T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 0.824T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 4.64T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 7.92T + 83T^{2} \) |
| 89 | \( 1 - 5.08T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74057331433690477468412394323, −7.57093805267821433964500542578, −6.70410873477823350476311959423, −5.59312241908603184930341612514, −4.56404430532087204965725299072, −3.81052667136475339508174272832, −3.09480298405574009484628650912, −2.31320489593063895883669126649, −1.54779433126547348028487444955, 0,
1.54779433126547348028487444955, 2.31320489593063895883669126649, 3.09480298405574009484628650912, 3.81052667136475339508174272832, 4.56404430532087204965725299072, 5.59312241908603184930341612514, 6.70410873477823350476311959423, 7.57093805267821433964500542578, 7.74057331433690477468412394323
