| L(s) = 1 | + 2.73·2-s + 5.46·4-s − 5-s + 9.46·8-s − 2.73·10-s − 0.732·11-s + 2.26·13-s + 14.9·16-s − 3.26·17-s + 4.46·19-s − 5.46·20-s − 2·22-s + 4.73·23-s + 25-s + 6.19·26-s + 4.19·29-s − 0.464·31-s + 21.8·32-s − 8.92·34-s − 3.19·37-s + 12.1·38-s − 9.46·40-s + 0.732·41-s + 3.19·43-s − 4·44-s + 12.9·46-s − 2·47-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s − 0.447·5-s + 3.34·8-s − 0.863·10-s − 0.220·11-s + 0.629·13-s + 3.73·16-s − 0.792·17-s + 1.02·19-s − 1.22·20-s − 0.426·22-s + 0.986·23-s + 0.200·25-s + 1.21·26-s + 0.779·29-s − 0.0833·31-s + 3.86·32-s − 1.53·34-s − 0.525·37-s + 1.97·38-s − 1.49·40-s + 0.114·41-s + 0.487·43-s − 0.603·44-s + 1.90·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.160550383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.160550383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 0.464T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 0.732T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.196T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.942765442895249633243179848159, −7.934143389558448207026804335539, −7.17006836423974039603091920964, −6.54098959762751938422260283484, −5.71365542842930466343236736977, −4.92676084689245965889564072181, −4.28490138845435264442344354248, −3.35453262065369419049874150601, −2.73384196480876704249664549308, −1.42328622967406263087918345191,
1.42328622967406263087918345191, 2.73384196480876704249664549308, 3.35453262065369419049874150601, 4.28490138845435264442344354248, 4.92676084689245965889564072181, 5.71365542842930466343236736977, 6.54098959762751938422260283484, 7.17006836423974039603091920964, 7.934143389558448207026804335539, 8.942765442895249633243179848159
