| L(s) = 1 | − 5.54·2-s + 7.25·3-s + 22.6·4-s − 1.00·5-s − 40.1·6-s + 21.1·7-s − 81.4·8-s + 25.6·9-s + 5.55·10-s − 6.83·11-s + 164.·12-s − 55.9·13-s − 117.·14-s − 7.27·15-s + 269.·16-s − 87.3·17-s − 141.·18-s + 79.2·19-s − 22.7·20-s + 153.·21-s + 37.8·22-s − 23·23-s − 590.·24-s − 123.·25-s + 309.·26-s − 9.99·27-s + 480.·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.39·3-s + 2.83·4-s − 0.0896·5-s − 2.73·6-s + 1.14·7-s − 3.59·8-s + 0.948·9-s + 0.175·10-s − 0.187·11-s + 3.96·12-s − 1.19·13-s − 2.23·14-s − 0.125·15-s + 4.21·16-s − 1.24·17-s − 1.85·18-s + 0.956·19-s − 0.254·20-s + 1.59·21-s + 0.366·22-s − 0.208·23-s − 5.02·24-s − 0.991·25-s + 2.33·26-s − 0.0712·27-s + 3.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 5.54T + 8T^{2} \) |
| 3 | \( 1 - 7.25T + 27T^{2} \) |
| 5 | \( 1 + 1.00T + 125T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 + 6.83T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.2T + 6.85e3T^{2} \) |
| 31 | \( 1 + 271.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 376.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 500.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 650.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 794.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 738.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.29T + 3.57e5T^{2} \) |
| 73 | \( 1 - 371.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 192.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 547.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 872.T + 9.12e5T^{2} \) |
| 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
−9.540238923636978326483547496897, −8.717552199663755295027184638989, −8.144078487505114142877098302677, −7.54423341970306838995042654716, −6.84111378040953133228613487874, −5.21392450891026374108902809542, −3.44965210271811856435115984386, −2.24454925047642522686068970454, −1.75633787832852061421627304473, 0,
1.75633787832852061421627304473, 2.24454925047642522686068970454, 3.44965210271811856435115984386, 5.21392450891026374108902809542, 6.84111378040953133228613487874, 7.54423341970306838995042654716, 8.144078487505114142877098302677, 8.717552199663755295027184638989, 9.540238923636978326483547496897
