| L(s) = 1 | + 2.10·2-s + 2.41·4-s + 0.560·5-s + 7-s + 0.873·8-s + 1.17·10-s − 5.80·11-s − 0.507·13-s + 2.10·14-s − 2.99·16-s + 0.446·17-s + 5.43·19-s + 1.35·20-s − 12.2·22-s + 5.14·23-s − 4.68·25-s − 1.06·26-s + 2.41·28-s + 7.22·29-s + 9.14·31-s − 8.04·32-s + 0.938·34-s + 0.560·35-s + 3.98·37-s + 11.4·38-s + 0.490·40-s − 4.97·41-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.20·4-s + 0.250·5-s + 0.377·7-s + 0.308·8-s + 0.372·10-s − 1.75·11-s − 0.140·13-s + 0.561·14-s − 0.748·16-s + 0.108·17-s + 1.24·19-s + 0.302·20-s − 2.60·22-s + 1.07·23-s − 0.937·25-s − 0.209·26-s + 0.456·28-s + 1.34·29-s + 1.64·31-s − 1.42·32-s + 0.161·34-s + 0.0947·35-s + 0.654·37-s + 1.85·38-s + 0.0774·40-s − 0.776·41-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
| \(L(1)\) |
\(\approx\) |
\(4.792041805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.792041805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 - 0.560T + 5T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 0.507T + 13T^{2} \) |
| 17 | \( 1 - 0.446T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 - 7.22T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 + 1.04T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 - 7.99T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−7.68877795807082726620583728475, −7.00139521072681654063175126722, −6.18482124076061158032390284440, −5.49571077760068170551234288788, −5.02101566155872761233050076920, −4.54595353117878351521186345367, −3.51084269193985671851277391980, −2.76985241878973125631113419016, −2.31577807302959132442359659124, −0.854878119358957291267611150551,
0.854878119358957291267611150551, 2.31577807302959132442359659124, 2.76985241878973125631113419016, 3.51084269193985671851277391980, 4.54595353117878351521186345367, 5.02101566155872761233050076920, 5.49571077760068170551234288788, 6.18482124076061158032390284440, 7.00139521072681654063175126722, 7.68877795807082726620583728475
