L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s + 2·13-s + 14-s − 15-s + 16-s − 6·17-s + 18-s − 4·19-s − 20-s + 21-s + 24-s + 25-s + 2·26-s + 27-s + 28-s − 6·29-s − 30-s − 4·31-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 + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.990424860\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.990424860\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.65179111469922693797071666945, −11.32821618685474578810063526343, −10.80914339692687256264302059978, −9.268646864684040562273313820184, −8.319627980737908137853866947983, −7.27965583167809929942184453108, −6.14620005069006563789988203374, −4.64849576847169223614699526425, −3.71822253537069739639390686069, −2.16105217917237072156284139202,
2.16105217917237072156284139202, 3.71822253537069739639390686069, 4.64849576847169223614699526425, 6.14620005069006563789988203374, 7.27965583167809929942184453108, 8.319627980737908137853866947983, 9.268646864684040562273313820184, 10.80914339692687256264302059978, 11.32821618685474578810063526343, 12.65179111469922693797071666945