| L(s) = 1 | − 2.01e10·3-s + 4.39e12·4-s − 8.21e14·5-s + 2.97e20·9-s − 7.40e21·11-s − 8.87e22·12-s + 1.65e25·15-s + 1.93e25·16-s − 3.61e27·20-s − 5.03e28·23-s + 4.47e29·25-s − 3.80e30·27-s − 3.16e31·31-s + 1.49e32·33-s + 1.30e33·36-s − 1.39e33·37-s − 3.25e34·44-s − 2.44e35·45-s − 1.84e35·47-s − 3.90e35·48-s + 3.11e35·49-s − 1.72e36·53-s + 6.07e36·55-s − 2.95e37·59-s + 7.29e37·60-s + 8.50e37·64-s − 2.98e38·67-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 4-s − 1.72·5-s + 2.72·9-s − 11-s − 1.92·12-s + 3.32·15-s + 16-s − 1.72·20-s − 1.27·23-s + 1.96·25-s − 3.32·27-s − 1.51·31-s + 1.92·33-s + 2.72·36-s − 1.63·37-s − 44-s − 4.68·45-s − 1.41·47-s − 1.92·48-s + 49-s − 1.06·53-s + 1.72·55-s − 1.91·59-s + 3.32·60-s + 64-s − 1.33·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{43}{2})\) |
\(\approx\) |
\(0.1616042128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1616042128\) |
| \(L(22)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{21} T \) |
| good | 2 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 3 | \( 1 + 20179560230 T + p^{42} T^{2} \) |
| 5 | \( 1 + 821522784070726 T + p^{42} T^{2} \) |
| 7 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 13 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 17 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 19 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 23 | \( 1 + \)\(50\!\cdots\!90\)\( T + p^{42} T^{2} \) |
| 29 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 31 | \( 1 + \)\(31\!\cdots\!62\)\( T + p^{42} T^{2} \) |
| 37 | \( 1 + \)\(13\!\cdots\!50\)\( T + p^{42} T^{2} \) |
| 41 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 43 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 47 | \( 1 + \)\(18\!\cdots\!50\)\( T + p^{42} T^{2} \) |
| 53 | \( 1 + \)\(17\!\cdots\!70\)\( T + p^{42} T^{2} \) |
| 59 | \( 1 + \)\(29\!\cdots\!18\)\( T + p^{42} T^{2} \) |
| 61 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 67 | \( 1 + \)\(29\!\cdots\!90\)\( T + p^{42} T^{2} \) |
| 71 | \( 1 + \)\(49\!\cdots\!58\)\( T + p^{42} T^{2} \) |
| 73 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 79 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 83 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 89 | \( 1 - \)\(77\!\cdots\!78\)\( T + p^{42} T^{2} \) |
| 97 | \( 1 + \)\(10\!\cdots\!30\)\( T + p^{42} T^{2} \) |
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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
−11.94920787130190621463992059814, −11.09490529369135406398275314634, −10.41791987809471171557634990382, −7.79452025240898737885074452739, −7.08090862042868938878965844112, −5.87982862784547956591800883053, −4.74759010333747104913657178123, −3.51906130900215202484316626597, −1.61706848927984707753110858382, −0.20478903873380430984247907408,
0.20478903873380430984247907408, 1.61706848927984707753110858382, 3.51906130900215202484316626597, 4.74759010333747104913657178123, 5.87982862784547956591800883053, 7.08090862042868938878965844112, 7.79452025240898737885074452739, 10.41791987809471171557634990382, 11.09490529369135406398275314634, 11.94920787130190621463992059814
