| L(s) = 1 | − 0.849·2-s − 1.27·4-s − 1.65·5-s + 7-s + 2.78·8-s + 1.40·10-s − 2.00·11-s − 2.46·13-s − 0.849·14-s + 0.190·16-s − 4.15·17-s − 5.98·19-s + 2.12·20-s + 1.70·22-s + 6.20·23-s − 2.24·25-s + 2.09·26-s − 1.27·28-s − 4.07·29-s − 6.85·31-s − 5.73·32-s + 3.52·34-s − 1.65·35-s − 9.80·37-s + 5.08·38-s − 4.62·40-s + 6.54·41-s + ⋯ |
| L(s) = 1 | − 0.600·2-s − 0.639·4-s − 0.741·5-s + 0.377·7-s + 0.984·8-s + 0.445·10-s − 0.605·11-s − 0.684·13-s − 0.227·14-s + 0.0477·16-s − 1.00·17-s − 1.37·19-s + 0.474·20-s + 0.363·22-s + 1.29·23-s − 0.449·25-s + 0.411·26-s − 0.241·28-s − 0.756·29-s − 1.23·31-s − 1.01·32-s + 0.604·34-s − 0.280·35-s − 1.61·37-s + 0.824·38-s − 0.730·40-s + 1.02·41-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
| \(L(1)\) |
\(\approx\) |
\(0.2784503188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2784503188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.849T + 2T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 + 5.98T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 6.94T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 + 5.28T + 61T^{2} \) |
| 67 | \( 1 - 0.780T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 0.329T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 6.36T + 97T^{2} \) |
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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
−7.87102224727902459637242829871, −7.33877615023680840467290493343, −6.77172368088613250280473662618, −5.54564875210305039280583823771, −4.99417692848154937952739665208, −4.26072713205081173025666289743, −3.72902463544996028333569801146, −2.51268019475293361804674778558, −1.66720911864615377492881028396, −0.28387965450738095360742542046,
0.28387965450738095360742542046, 1.66720911864615377492881028396, 2.51268019475293361804674778558, 3.72902463544996028333569801146, 4.26072713205081173025666289743, 4.99417692848154937952739665208, 5.54564875210305039280583823771, 6.77172368088613250280473662618, 7.33877615023680840467290493343, 7.87102224727902459637242829871
