| L(s) = 1 | + 2-s − 3-s + 4-s − 2.73·5-s − 6-s − 7-s + 8-s + 9-s − 2.73·10-s − 0.267·11-s − 12-s − 14-s + 2.73·15-s + 16-s + 3.73·17-s + 18-s − 2.46·19-s − 2.73·20-s + 21-s − 0.267·22-s − 3.46·23-s − 24-s + 2.46·25-s − 27-s − 28-s − 3·29-s + 2.73·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.22·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.863·10-s − 0.0807·11-s − 0.288·12-s − 0.267·14-s + 0.705·15-s + 0.250·16-s + 0.905·17-s + 0.235·18-s − 0.565·19-s − 0.610·20-s + 0.218·21-s − 0.0571·22-s − 0.722·23-s − 0.204·24-s + 0.492·25-s − 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 0.267T + 11T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 2.46T + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 + 0.928T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 0.464T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
| 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
−7.60491188952005286021430898257, −6.78944660457468836743577281520, −6.08929084243319816379316147130, −5.53776996053441100246173439024, −4.49083805521704451209780337618, −4.16418489296889987459682572543, −3.33888421009879985532914137220, −2.52133487055481954470454049275, −1.17294995297692128885344185238, 0,
1.17294995297692128885344185238, 2.52133487055481954470454049275, 3.33888421009879985532914137220, 4.16418489296889987459682572543, 4.49083805521704451209780337618, 5.53776996053441100246173439024, 6.08929084243319816379316147130, 6.78944660457468836743577281520, 7.60491188952005286021430898257
