L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s + 9-s + 10-s − 12-s + 13-s + 2·14-s − 15-s + 16-s + 18-s − 2·19-s + 20-s − 2·21-s − 6·23-s − 24-s + 25-s + 26-s − 27-s + 2·28-s − 30-s − 8·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.610611606\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.610611606\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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
−12.53655691165666, −12.11719556961891, −11.86284574050377, −11.19019988423187, −10.86593283776823, −10.54767001867718, −10.04695881412895, −9.369188925465856, −9.088314091277559, −8.318688215419657, −7.903267600646788, −7.542910161443893, −6.766812592463238, −6.531117967302948, −5.909891662041681, −5.553310882092847, −5.112285505808843, −4.551212958316799, −4.155295807625259, −3.565099115971012, −3.010780684321533, −2.195707120681386, −1.753027094132096, −1.389760214699373, −0.3633089135344463,
0.3633089135344463, 1.389760214699373, 1.753027094132096, 2.195707120681386, 3.010780684321533, 3.565099115971012, 4.155295807625259, 4.551212958316799, 5.112285505808843, 5.553310882092847, 5.909891662041681, 6.531117967302948, 6.766812592463238, 7.542910161443893, 7.903267600646788, 8.318688215419657, 9.088314091277559, 9.369188925465856, 10.04695881412895, 10.54767001867718, 10.86593283776823, 11.19019988423187, 11.86284574050377, 12.11719556961891, 12.53655691165666