| L(s) = 1 | − 1.13·2-s − 0.715·4-s − 1.11·5-s + 7-s + 3.07·8-s + 1.26·10-s − 3.25·11-s + 6.02·13-s − 1.13·14-s − 2.05·16-s − 2.98·17-s − 6.12·19-s + 0.795·20-s + 3.68·22-s − 5.74·23-s − 3.76·25-s − 6.83·26-s − 0.715·28-s − 7.65·29-s + 7.30·31-s − 3.82·32-s + 3.37·34-s − 1.11·35-s + 5.93·37-s + 6.93·38-s − 3.42·40-s + 4.23·41-s + ⋯ |
| L(s) = 1 | − 0.801·2-s − 0.357·4-s − 0.497·5-s + 0.377·7-s + 1.08·8-s + 0.398·10-s − 0.981·11-s + 1.67·13-s − 0.302·14-s − 0.514·16-s − 0.723·17-s − 1.40·19-s + 0.177·20-s + 0.786·22-s − 1.19·23-s − 0.752·25-s − 1.33·26-s − 0.135·28-s − 1.42·29-s + 1.31·31-s − 0.675·32-s + 0.579·34-s − 0.187·35-s + 0.974·37-s + 1.12·38-s − 0.540·40-s + 0.661·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6018029013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6018029013\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.13T + 2T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 0.455T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 - 4.44T + 83T^{2} \) |
| 89 | \( 1 - 6.16T + 89T^{2} \) |
| 97 | \( 1 - 3.77T + 97T^{2} \) |
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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
−7.918367261510159632242433679435, −7.59270260859117405065722077393, −6.42095826691959939229046876130, −5.91160160511631918085517205155, −4.92894469132216186229080918008, −4.13459817505525462981937964018, −3.80538707190684107517239587812, −2.41277193740247277728261322820, −1.62176675962362374230954633226, −0.44179024537702544246431051819,
0.44179024537702544246431051819, 1.62176675962362374230954633226, 2.41277193740247277728261322820, 3.80538707190684107517239587812, 4.13459817505525462981937964018, 4.92894469132216186229080918008, 5.91160160511631918085517205155, 6.42095826691959939229046876130, 7.59270260859117405065722077393, 7.918367261510159632242433679435
