L(s) = 1 | − 3-s − 2·5-s + 5·7-s + 9-s + 4·11-s − 5·13-s + 2·15-s − 5·21-s + 6·23-s − 25-s − 27-s − 8·29-s − 31-s − 4·33-s − 10·35-s − 7·37-s + 5·39-s + 11·43-s − 2·45-s − 10·47-s + 18·49-s − 6·53-s − 8·55-s + 8·59-s − 61-s + 5·63-s + 10·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.88·7-s + 1/3·9-s + 1.20·11-s − 1.38·13-s + 0.516·15-s − 1.09·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.696·33-s − 1.69·35-s − 1.15·37-s + 0.800·39-s + 1.67·43-s − 0.298·45-s − 1.45·47-s + 18/7·49-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 0.128·61-s + 0.629·63-s + 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.728232951\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.728232951\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.80408399385248, −15.14661829102152, −14.74817559294981, −14.44928285035827, −13.83991448713090, −12.87369887150527, −12.36697951408339, −11.81460014058233, −11.35936255270136, −11.15770675934158, −10.44960646287270, −9.558449137774814, −9.076971032391646, −8.340236222624898, −7.751689602412034, −7.249327816525636, −6.850163550220356, −5.755951782557686, −5.165026629782095, −4.648161138714521, −4.125369059807039, −3.387411894814970, −2.159219202577138, −1.551351636829185, −0.6010698326535333,
0.6010698326535333, 1.551351636829185, 2.159219202577138, 3.387411894814970, 4.125369059807039, 4.648161138714521, 5.165026629782095, 5.755951782557686, 6.850163550220356, 7.249327816525636, 7.751689602412034, 8.340236222624898, 9.076971032391646, 9.558449137774814, 10.44960646287270, 11.15770675934158, 11.35936255270136, 11.81460014058233, 12.36697951408339, 12.87369887150527, 13.83991448713090, 14.44928285035827, 14.74817559294981, 15.14661829102152, 15.80408399385248