| L(s) = 1 | + 1.37·3-s − 3.83·5-s − 1.10·9-s + 0.294·11-s − 2.83·13-s − 5.27·15-s − 3.23·17-s + 5.79·19-s + 5.90·23-s + 9.69·25-s − 5.65·27-s + 3.65·29-s + 1.59·31-s + 0.405·33-s + 5.20·37-s − 3.90·39-s + 41-s + 5.12·43-s + 4.22·45-s + 12.1·47-s − 4.46·51-s − 7.84·53-s − 1.12·55-s + 7.97·57-s − 2.50·59-s − 2.24·61-s + 10.8·65-s + ⋯ |
| L(s) = 1 | + 0.795·3-s − 1.71·5-s − 0.367·9-s + 0.0888·11-s − 0.786·13-s − 1.36·15-s − 0.785·17-s + 1.32·19-s + 1.23·23-s + 1.93·25-s − 1.08·27-s + 0.678·29-s + 0.285·31-s + 0.0706·33-s + 0.856·37-s − 0.625·39-s + 0.156·41-s + 0.780·43-s + 0.630·45-s + 1.77·47-s − 0.624·51-s − 1.07·53-s − 0.152·55-s + 1.05·57-s − 0.326·59-s − 0.287·61-s + 1.34·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 1.37T + 3T^{2} \) |
| 5 | \( 1 + 3.83T + 5T^{2} \) |
| 11 | \( 1 - 0.294T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 0.253T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.41629276906201980463261137994, −7.29182743346716527651950065365, −6.20322071249716567600498599089, −5.18966988893712392304271158222, −4.47739899588909342286505227119, −3.89361622210717576096438346922, −2.89834625647221771538531580482, −2.74795235403130939916995888225, −1.12458112653170182005938224774, 0,
1.12458112653170182005938224774, 2.74795235403130939916995888225, 2.89834625647221771538531580482, 3.89361622210717576096438346922, 4.47739899588909342286505227119, 5.18966988893712392304271158222, 6.20322071249716567600498599089, 7.29182743346716527651950065365, 7.41629276906201980463261137994
