| L(s) = 1 | + 5.31·2-s − 7.51·3-s + 20.2·4-s − 8.06·5-s − 39.9·6-s − 10.2·7-s + 64.9·8-s + 29.5·9-s − 42.8·10-s + 39.0·11-s − 151.·12-s + 21.0·13-s − 54.3·14-s + 60.6·15-s + 183.·16-s − 125.·17-s + 156.·18-s − 88.2·19-s − 163.·20-s + 76.9·21-s + 207.·22-s + 23·23-s − 487.·24-s − 59.9·25-s + 111.·26-s − 18.9·27-s − 206.·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 1.44·3-s + 2.52·4-s − 0.721·5-s − 2.71·6-s − 0.552·7-s + 2.86·8-s + 1.09·9-s − 1.35·10-s + 1.07·11-s − 3.65·12-s + 0.448·13-s − 1.03·14-s + 1.04·15-s + 2.85·16-s − 1.79·17-s + 2.05·18-s − 1.06·19-s − 1.82·20-s + 0.799·21-s + 2.00·22-s + 0.208·23-s − 4.14·24-s − 0.479·25-s + 0.841·26-s − 0.135·27-s − 1.39·28-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 | \( 1 - 5.31T + 8T^{2} \) |
| 3 | \( 1 + 7.51T + 27T^{2} \) |
| 5 | \( 1 + 8.06T + 125T^{2} \) |
| 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.2T + 6.85e3T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 37.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 841.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 638.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 66.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 890.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 297.T + 9.12e5T^{2} \) |
| 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
−10.28692651288867133880448172234, −8.739365142493145294445365764013, −7.18136855069972435093505365034, −6.48194125662977760101146181567, −6.12665058167672929794262813846, −4.95961976473557724666529835180, −4.23631483887554370592088629632, −3.49834926277943117553150937094, −1.85571937801649154188772963552, 0,
1.85571937801649154188772963552, 3.49834926277943117553150937094, 4.23631483887554370592088629632, 4.95961976473557724666529835180, 6.12665058167672929794262813846, 6.48194125662977760101146181567, 7.18136855069972435093505365034, 8.739365142493145294445365764013, 10.28692651288867133880448172234
