L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 2·11-s + 12-s + 13-s + 14-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s + 21-s + 2·22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.07994857091434, −12.44651132659504, −11.86319723384639, −11.59766378308502, −11.09615813904766, −10.66214422993380, −10.07371920024351, −9.530961833070760, −9.302342079864233, −8.693793154101124, −8.229692095606365, −7.677040771188120, −7.275729373268207, −6.736491630145557, −6.249432931750399, −5.853415222997593, −5.143286557068877, −4.810340235631026, −4.326948145587147, −3.567233098661925, −3.329302105637843, −2.763631998607047, −1.952777831879319, −1.664019257579570, −1.093947547081615, 0,
1.093947547081615, 1.664019257579570, 1.952777831879319, 2.763631998607047, 3.329302105637843, 3.567233098661925, 4.326948145587147, 4.810340235631026, 5.143286557068877, 5.853415222997593, 6.249432931750399, 6.736491630145557, 7.275729373268207, 7.677040771188120, 8.229692095606365, 8.693793154101124, 9.302342079864233, 9.530961833070760, 10.07371920024351, 10.66214422993380, 11.09615813904766, 11.59766378308502, 11.86319723384639, 12.44651132659504, 13.07994857091434