| L(s) = 1 | − 3.82·2-s − 3·3-s + 6.65·4-s + 5·5-s + 11.4·6-s − 7·7-s + 5.14·8-s + 9·9-s − 19.1·10-s + 48.5·11-s − 19.9·12-s − 43.6·13-s + 26.7·14-s − 15·15-s − 72.9·16-s − 67.6·17-s − 34.4·18-s − 93.2·19-s + 33.2·20-s + 21·21-s − 185.·22-s − 104.·23-s − 15.4·24-s + 25·25-s + 167.·26-s − 27·27-s − 46.5·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.577·3-s + 0.832·4-s + 0.447·5-s + 0.781·6-s − 0.377·7-s + 0.227·8-s + 0.333·9-s − 0.605·10-s + 1.33·11-s − 0.480·12-s − 0.931·13-s + 0.511·14-s − 0.258·15-s − 1.13·16-s − 0.965·17-s − 0.451·18-s − 1.12·19-s + 0.372·20-s + 0.218·21-s − 1.80·22-s − 0.944·23-s − 0.131·24-s + 0.200·25-s + 1.26·26-s − 0.192·27-s − 0.314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 11 | \( 1 - 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.08T + 2.97e4T^{2} \) |
| 37 | \( 1 + 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 529.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 463.T + 9.12e5T^{2} \) |
| 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
−12.52362329578303523174612218328, −11.40906351113754730786648519606, −10.35731378291167851175667760938, −9.522381603063368539181413992794, −8.655846868553477110650215124948, −7.15431105242583476581047669936, −6.25238828112403108123191162025, −4.41405209067270110225811331286, −1.85312014652769458321998380111, 0,
1.85312014652769458321998380111, 4.41405209067270110225811331286, 6.25238828112403108123191162025, 7.15431105242583476581047669936, 8.655846868553477110650215124948, 9.522381603063368539181413992794, 10.35731378291167851175667760938, 11.40906351113754730786648519606, 12.52362329578303523174612218328
