| L(s) = 1 | + 0.614·5-s − 3.46·7-s − 3.80·11-s − 2.93·13-s − 3.15·17-s − 5.58·19-s + 0.574·23-s − 4.62·25-s + 3.25·29-s − 1.74·31-s − 2.12·35-s − 4.35·37-s + 1.51·41-s + 9.97·43-s + 8.29·47-s + 4.99·49-s − 0.465·53-s − 2.34·55-s + 9.58·59-s − 6.22·61-s − 1.80·65-s + 3.41·67-s + 8.58·71-s − 9.90·73-s + 13.1·77-s + 10.2·79-s + 8.39·83-s + ⋯ |
| L(s) = 1 | + 0.275·5-s − 1.30·7-s − 1.14·11-s − 0.812·13-s − 0.765·17-s − 1.28·19-s + 0.119·23-s − 0.924·25-s + 0.603·29-s − 0.313·31-s − 0.359·35-s − 0.716·37-s + 0.236·41-s + 1.52·43-s + 1.20·47-s + 0.713·49-s − 0.0638·53-s − 0.315·55-s + 1.24·59-s − 0.797·61-s − 0.223·65-s + 0.416·67-s + 1.01·71-s − 1.15·73-s + 1.50·77-s + 1.15·79-s + 0.921·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7690382344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7690382344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.614T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 - 0.574T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 + 0.465T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 8.39T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.015691554841912529144819243680, −7.32710114399640974803504918691, −6.61464871371783242190414027291, −6.02407451502591836181516925387, −5.28196737942227455087274558946, −4.42759416258627250006456211627, −3.62510106029962839840631464629, −2.58105470431620863279746074866, −2.20414976932895680863978180055, −0.42439600878059316446815568226,
0.42439600878059316446815568226, 2.20414976932895680863978180055, 2.58105470431620863279746074866, 3.62510106029962839840631464629, 4.42759416258627250006456211627, 5.28196737942227455087274558946, 6.02407451502591836181516925387, 6.61464871371783242190414027291, 7.32710114399640974803504918691, 8.015691554841912529144819243680
