| L(s) = 1 | − 5·3-s + 4·4-s − 5-s + 16·9-s − 11·11-s − 20·12-s + 5·15-s + 16·16-s − 4·20-s + 35·23-s − 24·25-s − 35·27-s − 37·31-s + 55·33-s + 64·36-s − 25·37-s − 44·44-s − 16·45-s + 50·47-s − 80·48-s + 49·49-s − 70·53-s + 11·55-s + 107·59-s + 20·60-s + 64·64-s + 35·67-s + ⋯ |
| L(s) = 1 | − 5/3·3-s + 4-s − 1/5·5-s + 16/9·9-s − 11-s − 5/3·12-s + 1/3·15-s + 16-s − 1/5·20-s + 1.52·23-s − 0.959·25-s − 1.29·27-s − 1.19·31-s + 5/3·33-s + 16/9·36-s − 0.675·37-s − 44-s − 0.355·45-s + 1.06·47-s − 5/3·48-s + 49-s − 1.32·53-s + 1/5·55-s + 1.81·59-s + 1/3·60-s + 64-s + 0.522·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5529457047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5529457047\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( 1 + 5 T + p^{2} T^{2} \) |
| 5 | \( 1 + T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 35 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 37 T + p^{2} T^{2} \) |
| 37 | \( 1 + 25 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 50 T + p^{2} T^{2} \) |
| 53 | \( 1 + 70 T + p^{2} T^{2} \) |
| 59 | \( 1 - 107 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 35 T + p^{2} T^{2} \) |
| 71 | \( 1 + 133 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 97 T + p^{2} T^{2} \) |
| 97 | \( 1 - 95 T + p^{2} T^{2} \) |
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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
−20.72706949828071681393765289837, −18.94937433851464141337121078848, −17.50931498854494012695157990131, −16.39332234475921287320684070443, −15.41095432208513402598326186997, −12.72041557714426109759504904958, −11.45633132707164803753858409043, −10.50725023499595595838468875295, −7.18564016333659252457943769445, −5.54693185652010333576723147591,
5.54693185652010333576723147591, 7.18564016333659252457943769445, 10.50725023499595595838468875295, 11.45633132707164803753858409043, 12.72041557714426109759504904958, 15.41095432208513402598326186997, 16.39332234475921287320684070443, 17.50931498854494012695157990131, 18.94937433851464141337121078848, 20.72706949828071681393765289837
