| L(s) = 1 | + 3-s + 4·5-s + 7-s + 9-s + 13-s + 4·15-s − 17-s + 2·19-s + 21-s + 4·23-s + 11·25-s + 27-s − 29-s − 11·31-s + 4·35-s + 3·37-s + 39-s − 6·41-s − 3·43-s + 4·45-s − 9·47-s − 6·49-s − 51-s − 2·53-s + 2·57-s − 9·59-s + 12·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s + 1.03·15-s − 0.242·17-s + 0.458·19-s + 0.218·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 0.185·29-s − 1.97·31-s + 0.676·35-s + 0.493·37-s + 0.160·39-s − 0.937·41-s − 0.457·43-s + 0.596·45-s − 1.31·47-s − 6/7·49-s − 0.140·51-s − 0.274·53-s + 0.264·57-s − 1.17·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.96708828043227, −12.76355934175619, −11.92703255917349, −11.33605048575227, −11.05167253307279, −10.43271226490260, −10.06263278287038, −9.609436871475098, −9.219014638524730, −8.782540384012605, −8.496592168175742, −7.660943956630998, −7.377432855726463, −6.699616684888043, −6.327559934422580, −5.849069214829047, −5.150653742455150, −5.050961913417431, −4.389397581192622, −3.411927156011473, −3.309541176811546, −2.491806010802484, −2.005916396555064, −1.549134770719489, −1.130767234639261, 0,
1.130767234639261, 1.549134770719489, 2.005916396555064, 2.491806010802484, 3.309541176811546, 3.411927156011473, 4.389397581192622, 5.050961913417431, 5.150653742455150, 5.849069214829047, 6.327559934422580, 6.699616684888043, 7.377432855726463, 7.660943956630998, 8.496592168175742, 8.782540384012605, 9.219014638524730, 9.609436871475098, 10.06263278287038, 10.43271226490260, 11.05167253307279, 11.33605048575227, 11.92703255917349, 12.76355934175619, 12.96708828043227
