| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s + 2·11-s − 12-s − 6·13-s + 4·14-s − 15-s + 16-s + 18-s − 2·19-s + 20-s − 4·21-s + 2·22-s + 4·23-s − 24-s + 25-s − 6·26-s − 27-s + 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 1.66·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.872·21-s + 0.426·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.64571485135021, −12.26587245308924, −11.80080716396520, −11.47741907617648, −11.03018580818336, −10.56105572741045, −10.21082928001774, −9.600907146300171, −9.121605944805471, −8.635966417067078, −8.017283567858397, −7.557708848100149, −7.124234665175906, −6.657378415610400, −6.280521451336744, −5.391095392791980, −5.199459123323018, −4.961770135185455, −4.366094294599473, −3.887137354329356, −3.193860494291833, −2.505201996989121, −1.919752645403170, −1.648709665245420, −0.8987559856955546, 0,
0.8987559856955546, 1.648709665245420, 1.919752645403170, 2.505201996989121, 3.193860494291833, 3.887137354329356, 4.366094294599473, 4.961770135185455, 5.199459123323018, 5.391095392791980, 6.280521451336744, 6.657378415610400, 7.124234665175906, 7.557708848100149, 8.017283567858397, 8.635966417067078, 9.121605944805471, 9.600907146300171, 10.21082928001774, 10.56105572741045, 11.03018580818336, 11.47741907617648, 11.80080716396520, 12.26587245308924, 12.64571485135021
