| L(s) = 1 | + 0.536·3-s − 3.30·5-s − 7-s − 2.71·9-s + 11-s + 13-s − 1.77·15-s + 3.13·17-s − 3.06·19-s − 0.536·21-s + 0.420·23-s + 5.89·25-s − 3.06·27-s + 0.161·29-s + 10.3·31-s + 0.536·33-s + 3.30·35-s + 4.54·37-s + 0.536·39-s − 9.30·41-s + 2.24·43-s + 8.94·45-s + 2.05·47-s + 49-s + 1.68·51-s + 8.66·53-s − 3.30·55-s + ⋯ |
| L(s) = 1 | + 0.309·3-s − 1.47·5-s − 0.377·7-s − 0.903·9-s + 0.301·11-s + 0.277·13-s − 0.457·15-s + 0.759·17-s − 0.702·19-s − 0.117·21-s + 0.0876·23-s + 1.17·25-s − 0.589·27-s + 0.0300·29-s + 1.86·31-s + 0.0934·33-s + 0.557·35-s + 0.747·37-s + 0.0859·39-s − 1.45·41-s + 0.342·43-s + 1.33·45-s + 0.300·47-s + 0.142·49-s + 0.235·51-s + 1.19·53-s − 0.444·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.536T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 0.420T + 23T^{2} \) |
| 29 | \( 1 - 0.161T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 4.62T + 73T^{2} \) |
| 79 | \( 1 + 6.36T + 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 - 5.49T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.66841611195648886392753834310, −6.83586040696135180162873642541, −6.21819276240802729905519894930, −5.38621689705733445270247381610, −4.45192031708981803783745479257, −3.84442129392224333557781188531, −3.18375683905111387990470460836, −2.50515502191842432925008762267, −1.04709616696701113969506369400, 0,
1.04709616696701113969506369400, 2.50515502191842432925008762267, 3.18375683905111387990470460836, 3.84442129392224333557781188531, 4.45192031708981803783745479257, 5.38621689705733445270247381610, 6.21819276240802729905519894930, 6.83586040696135180162873642541, 7.66841611195648886392753834310
