L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 14-s − 15-s + 16-s + 4·17-s − 18-s − 20-s − 21-s + 22-s + 2·23-s − 24-s + 25-s + 27-s − 28-s + 6·29-s + 30-s − 8·31-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 − 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.340164658\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340164658\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.751267889800360478331772481796, −8.499638386230823686054482580356, −7.33067230791885941556327877079, −7.19615988029899304661087471318, −5.97635346909371001203977353417, −5.09439904526314630426457361956, −3.85623906214277198746293646442, −3.15261208746659791150030854340, −2.14806682404755852758870731479, −0.807755251224935336667185648653,
0.807755251224935336667185648653, 2.14806682404755852758870731479, 3.15261208746659791150030854340, 3.85623906214277198746293646442, 5.09439904526314630426457361956, 5.97635346909371001203977353417, 7.19615988029899304661087471318, 7.33067230791885941556327877079, 8.499638386230823686054482580356, 8.751267889800360478331772481796