| L(s) = 1 | − 2.18·2-s − 1.75·3-s + 2.76·4-s + 2.11·5-s + 3.83·6-s − 1.67·8-s + 0.0920·9-s − 4.60·10-s + 5.76·11-s − 4.86·12-s − 3.71·15-s − 1.87·16-s − 1.64·17-s − 0.200·18-s − 2.67·19-s + 5.83·20-s − 12.5·22-s + 6.42·23-s + 2.94·24-s − 0.545·25-s + 5.11·27-s − 6.04·29-s + 8.10·30-s + 5.12·31-s + 7.45·32-s − 10.1·33-s + 3.58·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 1.01·3-s + 1.38·4-s + 0.943·5-s + 1.56·6-s − 0.591·8-s + 0.0306·9-s − 1.45·10-s + 1.73·11-s − 1.40·12-s − 0.958·15-s − 0.469·16-s − 0.397·17-s − 0.0473·18-s − 0.612·19-s + 1.30·20-s − 2.68·22-s + 1.33·23-s + 0.600·24-s − 0.109·25-s + 0.984·27-s − 1.12·29-s + 1.47·30-s + 0.919·31-s + 1.31·32-s − 1.76·33-s + 0.614·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 2.11T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 - 1.14T + 89T^{2} \) |
| 97 | \( 1 + 6.97T + 97T^{2} \) |
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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
−7.29320335341306187993603698124, −6.82652719939054666553359279354, −6.29717149076399793938666343610, −5.68181698395294707755949662987, −4.82602986567969773478438962742, −3.93827628833901261994868446585, −2.68517917400007787453736319583, −1.69427411284801069372101919041, −1.13165733680551806191800049742, 0,
1.13165733680551806191800049742, 1.69427411284801069372101919041, 2.68517917400007787453736319583, 3.93827628833901261994868446585, 4.82602986567969773478438962742, 5.68181698395294707755949662987, 6.29717149076399793938666343610, 6.82652719939054666553359279354, 7.29320335341306187993603698124
