| L(s) = 1 | − 0.0449·2-s − 1.99·4-s + 2.40·5-s + 7-s + 0.179·8-s − 0.108·10-s + 3.19·11-s + 3.02·13-s − 0.0449·14-s + 3.98·16-s + 2.65·17-s − 1.70·19-s − 4.80·20-s − 0.143·22-s + 5.96·23-s + 0.787·25-s − 0.135·26-s − 1.99·28-s − 4.60·29-s − 0.446·31-s − 0.539·32-s − 0.119·34-s + 2.40·35-s + 2.06·37-s + 0.0766·38-s + 0.432·40-s + 7.56·41-s + ⋯ |
| L(s) = 1 | − 0.0318·2-s − 0.998·4-s + 1.07·5-s + 0.377·7-s + 0.0636·8-s − 0.0342·10-s + 0.962·11-s + 0.838·13-s − 0.0120·14-s + 0.996·16-s + 0.645·17-s − 0.390·19-s − 1.07·20-s − 0.0306·22-s + 1.24·23-s + 0.157·25-s − 0.0266·26-s − 0.377·28-s − 0.855·29-s − 0.0801·31-s − 0.0953·32-s − 0.0205·34-s + 0.406·35-s + 0.339·37-s + 0.0124·38-s + 0.0684·40-s + 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.581897328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.581897328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0449T + 2T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 + 0.446T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 - 9.90T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 + 0.470T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 6.16T + 97T^{2} \) |
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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
−7.88316019296902108293097166328, −7.21457193805106118551306996803, −6.15935246055760667147675045123, −5.82221628335395764360897721051, −5.10567807527166312605478685226, −4.24369381143079088113290722207, −3.70224353259428419600760773674, −2.64996181246509873580991960025, −1.53565571306957814264478150382, −0.907446849896959862177254653810,
0.907446849896959862177254653810, 1.53565571306957814264478150382, 2.64996181246509873580991960025, 3.70224353259428419600760773674, 4.24369381143079088113290722207, 5.10567807527166312605478685226, 5.82221628335395764360897721051, 6.15935246055760667147675045123, 7.21457193805106118551306996803, 7.88316019296902108293097166328
