| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 7-s + 8-s + 9-s − 3·10-s − 4·11-s − 12-s − 3·13-s + 14-s + 3·15-s + 16-s + 18-s − 3·20-s − 21-s − 4·22-s − 23-s − 24-s + 4·25-s − 3·26-s − 27-s + 28-s − 3·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.235·18-s − 0.670·20-s − 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.188·28-s − 0.557·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−12.80345371994606, −12.54237617024390, −12.02270158724558, −11.55873645593232, −11.41633766214720, −10.81754226025148, −10.42989458223143, −9.836522160610011, −9.514075476203682, −8.526421683409733, −8.122686457866956, −7.805591501635984, −7.441213024504779, −6.789539889627695, −6.466677162697700, −5.717705880284902, −5.161810924332273, −4.941819548898576, −4.324508763473684, −4.023330341071839, −3.230652429168381, −2.850514118883806, −2.188730427589246, −1.490018917219294, −0.5856539203236870, 0,
0.5856539203236870, 1.490018917219294, 2.188730427589246, 2.850514118883806, 3.230652429168381, 4.023330341071839, 4.324508763473684, 4.941819548898576, 5.161810924332273, 5.717705880284902, 6.466677162697700, 6.789539889627695, 7.441213024504779, 7.805591501635984, 8.122686457866956, 8.526421683409733, 9.514075476203682, 9.836522160610011, 10.42989458223143, 10.81754226025148, 11.41633766214720, 11.55873645593232, 12.02270158724558, 12.54237617024390, 12.80345371994606
