L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s + 4·7-s − 8-s + 9-s − 4·10-s − 11-s + 12-s − 4·14-s + 4·15-s + 16-s − 17-s − 18-s + 2·19-s + 4·20-s + 4·21-s + 22-s + 2·23-s − 24-s + 11·25-s + 27-s + 4·28-s − 10·29-s − 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s − 1.06·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.894·20-s + 0.872·21-s + 0.213·22-s + 0.417·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.755·28-s − 1.85·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.189811201\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.189811201\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.26332059249001, −12.81514527754813, −12.17162611209236, −11.39756035197168, −11.17550162819556, −10.74215915242792, −10.11236658021855, −9.712591226595669, −9.401424912834087, −8.816122937995696, −8.516676017186313, −7.811584152754227, −7.590439873342811, −6.898265723959818, −6.395353811644015, −5.785717631652730, −5.270449875623116, −4.983660285954212, −4.267654511502724, −3.456651825842724, −2.772387209583391, −2.239888625670131, −1.765097735736348, −1.462202237940304, −0.6787763736755506,
0.6787763736755506, 1.462202237940304, 1.765097735736348, 2.239888625670131, 2.772387209583391, 3.456651825842724, 4.267654511502724, 4.983660285954212, 5.270449875623116, 5.785717631652730, 6.395353811644015, 6.898265723959818, 7.590439873342811, 7.811584152754227, 8.516676017186313, 8.816122937995696, 9.401424912834087, 9.712591226595669, 10.11236658021855, 10.74215915242792, 11.17550162819556, 11.39756035197168, 12.17162611209236, 12.81514527754813, 13.26332059249001