| L(s) = 1 | + 4.96·2-s + 0.827·3-s + 16.6·4-s − 20.5·5-s + 4.11·6-s + 11.3·7-s + 43.1·8-s − 26.3·9-s − 102.·10-s − 30.8·11-s + 13.8·12-s − 12.3·13-s + 56.3·14-s − 17.0·15-s + 81.0·16-s − 33.6·17-s − 130.·18-s − 102.·19-s − 343.·20-s + 9.37·21-s − 153.·22-s + 35.7·24-s + 298.·25-s − 61.5·26-s − 44.1·27-s + 189.·28-s + 68.9·29-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.159·3-s + 2.08·4-s − 1.84·5-s + 0.279·6-s + 0.611·7-s + 1.90·8-s − 0.974·9-s − 3.23·10-s − 0.845·11-s + 0.332·12-s − 0.264·13-s + 1.07·14-s − 0.293·15-s + 1.26·16-s − 0.480·17-s − 1.71·18-s − 1.23·19-s − 3.84·20-s + 0.0974·21-s − 1.48·22-s + 0.303·24-s + 2.39·25-s − 0.464·26-s − 0.314·27-s + 1.27·28-s + 0.441·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 - 4.96T + 8T^{2} \) |
| 3 | \( 1 - 0.827T + 27T^{2} \) |
| 5 | \( 1 + 20.5T + 125T^{2} \) |
| 7 | \( 1 - 11.3T + 343T^{2} \) |
| 11 | \( 1 + 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 68.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 66.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 66.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 498.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 439.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 499.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 583.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 30.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 268.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 364.T + 7.04e5T^{2} \) |
| 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
−10.71411041251918786858002556447, −8.708695728173674029749131201747, −7.950865138968795483257656145423, −7.19913131466413898111130095122, −6.04542057921851398841443210927, −4.90647579189758657702030698921, −4.31349646235203894761785750832, −3.33653197588260019681324849243, −2.40264190465050538491655862453, 0,
2.40264190465050538491655862453, 3.33653197588260019681324849243, 4.31349646235203894761785750832, 4.90647579189758657702030698921, 6.04542057921851398841443210927, 7.19913131466413898111130095122, 7.950865138968795483257656145423, 8.708695728173674029749131201747, 10.71411041251918786858002556447
