L(s) = 1 | − 2-s − 1.44·3-s + 4-s − 2.24·5-s + 1.44·6-s − 7-s − 8-s − 0.911·9-s + 2.24·10-s + 2.19·11-s − 1.44·12-s + 14-s + 3.24·15-s + 16-s − 8.00·17-s + 0.911·18-s + 4.44·19-s − 2.24·20-s + 1.44·21-s − 2.19·22-s − 4.13·23-s + 1.44·24-s + 0.0489·25-s + 5.65·27-s − 28-s − 8.45·29-s − 3.24·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.834·3-s + 0.5·4-s − 1.00·5-s + 0.589·6-s − 0.377·7-s − 0.353·8-s − 0.303·9-s + 0.710·10-s + 0.662·11-s − 0.417·12-s + 0.267·14-s + 0.838·15-s + 0.250·16-s − 1.94·17-s + 0.214·18-s + 1.01·19-s − 0.502·20-s + 0.315·21-s − 0.468·22-s − 0.862·23-s + 0.294·24-s + 0.00978·25-s + 1.08·27-s − 0.188·28-s − 1.57·29-s − 0.592·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2918131775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2918131775\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 17 | \( 1 + 8.00T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 - 0.472T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 + 0.0217T + 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 3.15T + 73T^{2} \) |
| 79 | \( 1 - 2.77T + 79T^{2} \) |
| 83 | \( 1 + 1.91T + 83T^{2} \) |
| 89 | \( 1 - 9.72T + 89T^{2} \) |
| 97 | \( 1 + 4.05T + 97T^{2} \) |
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
−8.841274145577112149541777396775, −8.367623652806729213259873683704, −7.35374733161056425724395300956, −6.75032211671209814742780890317, −6.06744560151831436886881709484, −5.10289504619940791190326633301, −4.09317204128127398537116563886, −3.25308847086375744687264018743, −1.91525654954894085705496731210, −0.38560701585570406103840459407,
0.38560701585570406103840459407, 1.91525654954894085705496731210, 3.25308847086375744687264018743, 4.09317204128127398537116563886, 5.10289504619940791190326633301, 6.06744560151831436886881709484, 6.75032211671209814742780890317, 7.35374733161056425724395300956, 8.367623652806729213259873683704, 8.841274145577112149541777396775