| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 15-s + 16-s − 17-s − 18-s − 2·19-s − 20-s + 24-s + 25-s − 27-s − 8·29-s − 30-s + 8·31-s − 32-s + 34-s + 36-s − 6·37-s + 2·38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.986·37-s + 0.324·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.52914530993260, −13.87211654566034, −13.34892735469231, −12.79500422447931, −12.33632573066227, −11.69913901067262, −11.55719760146651, −10.79307268159452, −10.56590230448742, −9.972937345338392, −9.361048397891153, −9.000482213082101, −8.234295090452474, −7.947698365760308, −7.372430050415940, −6.621677318853153, −6.503198083742488, −5.745040837841969, −5.079022110961157, −4.611158213064460, −3.887669787593321, −3.280224521345589, −2.615538995695593, −1.707381547516027, −1.302635991986334, 0, 0,
1.302635991986334, 1.707381547516027, 2.615538995695593, 3.280224521345589, 3.887669787593321, 4.611158213064460, 5.079022110961157, 5.745040837841969, 6.503198083742488, 6.621677318853153, 7.372430050415940, 7.947698365760308, 8.234295090452474, 9.000482213082101, 9.361048397891153, 9.972937345338392, 10.56590230448742, 10.79307268159452, 11.55719760146651, 11.69913901067262, 12.33632573066227, 12.79500422447931, 13.34892735469231, 13.87211654566034, 14.52914530993260
