| L(s) = 1 | − 1.56·3-s + 5-s + 2.56·7-s − 0.561·9-s + 2·11-s − 3.56·13-s − 1.56·15-s + 2.56·17-s + 6·19-s − 4·21-s + 23-s + 25-s + 5.56·27-s + 6.12·29-s + 7.24·31-s − 3.12·33-s + 2.56·35-s − 4.56·37-s + 5.56·39-s + 4.12·41-s − 0.561·45-s + 4.68·47-s − 0.438·49-s − 4·51-s − 4.56·53-s + 2·55-s − 9.36·57-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s + 0.968·7-s − 0.187·9-s + 0.603·11-s − 0.987·13-s − 0.403·15-s + 0.621·17-s + 1.37·19-s − 0.872·21-s + 0.208·23-s + 0.200·25-s + 1.07·27-s + 1.13·29-s + 1.30·31-s − 0.543·33-s + 0.432·35-s − 0.749·37-s + 0.890·39-s + 0.643·41-s − 0.0837·45-s + 0.683·47-s − 0.0626·49-s − 0.560·51-s − 0.626·53-s + 0.269·55-s − 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.241369082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241369082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 + 8.56T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−11.15705410614823363849934995922, −10.26069663919525500049489331790, −9.413567443793721788312754260038, −8.296926865390312795284774797651, −7.30908675113184739361458820548, −6.24974283601122405255968479225, −5.30024816944262028073283923701, −4.64545925771980260631492864359, −2.87681778251835371546309168946, −1.19389131664706820276214767776,
1.19389131664706820276214767776, 2.87681778251835371546309168946, 4.64545925771980260631492864359, 5.30024816944262028073283923701, 6.24974283601122405255968479225, 7.30908675113184739361458820548, 8.296926865390312795284774797651, 9.413567443793721788312754260038, 10.26069663919525500049489331790, 11.15705410614823363849934995922
