| L(s) = 1 | + 2.06·2-s − 3-s + 2.27·4-s − 0.347·5-s − 2.06·6-s − 4.33·7-s + 0.558·8-s + 9-s − 0.718·10-s − 0.729·11-s − 2.27·12-s − 5.40·13-s − 8.96·14-s + 0.347·15-s − 3.38·16-s + 2.06·18-s + 3.27·19-s − 0.789·20-s + 4.33·21-s − 1.50·22-s + 3.55·23-s − 0.558·24-s − 4.87·25-s − 11.1·26-s − 27-s − 9.84·28-s − 4.38·29-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 0.577·3-s + 1.13·4-s − 0.155·5-s − 0.843·6-s − 1.63·7-s + 0.197·8-s + 0.333·9-s − 0.227·10-s − 0.220·11-s − 0.655·12-s − 1.49·13-s − 2.39·14-s + 0.0897·15-s − 0.846·16-s + 0.487·18-s + 0.750·19-s − 0.176·20-s + 0.946·21-s − 0.321·22-s + 0.741·23-s − 0.113·24-s − 0.975·25-s − 2.18·26-s − 0.192·27-s − 1.86·28-s − 0.813·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.06T + 2T^{2} \) |
| 5 | \( 1 + 0.347T + 5T^{2} \) |
| 7 | \( 1 + 4.33T + 7T^{2} \) |
| 11 | \( 1 + 0.729T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 - 0.476T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 0.812T + 79T^{2} \) |
| 83 | \( 1 - 7.21T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−9.697427215451327773979804131940, −9.305585940269370712730658202208, −7.49190462221568915860355072734, −6.92898191325120643392909360978, −5.94469233384689415236144386455, −5.37682354622876460132835452223, −4.35003063498592884708010282475, −3.40835569048687504232128171461, −2.52892882738560049276794870196, 0,
2.52892882738560049276794870196, 3.40835569048687504232128171461, 4.35003063498592884708010282475, 5.37682354622876460132835452223, 5.94469233384689415236144386455, 6.92898191325120643392909360978, 7.49190462221568915860355072734, 9.305585940269370712730658202208, 9.697427215451327773979804131940
