| L(s) = 1 | + 3.18·3-s − 0.942·5-s − 1.31·7-s + 7.15·9-s + 1.39·11-s + 4.30·13-s − 3.00·15-s + 3.89·17-s − 19-s − 4.19·21-s − 0.421·23-s − 4.11·25-s + 13.2·27-s + 7.00·29-s + 6.02·31-s + 4.44·33-s + 1.24·35-s − 1.80·37-s + 13.7·39-s − 5.83·41-s − 2.59·43-s − 6.74·45-s + 2.39·47-s − 5.26·49-s + 12.4·51-s − 5.08·53-s − 1.31·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 0.421·5-s − 0.497·7-s + 2.38·9-s + 0.420·11-s + 1.19·13-s − 0.775·15-s + 0.944·17-s − 0.229·19-s − 0.915·21-s − 0.0879·23-s − 0.822·25-s + 2.54·27-s + 1.29·29-s + 1.08·31-s + 0.773·33-s + 0.209·35-s − 0.297·37-s + 2.19·39-s − 0.911·41-s − 0.395·43-s − 1.00·45-s + 0.348·47-s − 0.752·49-s + 1.73·51-s − 0.698·53-s − 0.177·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.211576305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.211576305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 0.942T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 2.39T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.985733680444960111078112119488, −7.82598283790655843662026147833, −6.66695450977577502250037645800, −6.30834465581343851195018663524, −4.96163516372317320642284063536, −4.06636049498906107896924932949, −3.48647929858038756679431980630, −3.03458525933877607331909540838, −1.97691068010108121297380284249, −1.05822778233550536773401930665,
1.05822778233550536773401930665, 1.97691068010108121297380284249, 3.03458525933877607331909540838, 3.48647929858038756679431980630, 4.06636049498906107896924932949, 4.96163516372317320642284063536, 6.30834465581343851195018663524, 6.66695450977577502250037645800, 7.82598283790655843662026147833, 7.985733680444960111078112119488
