L(s) = 1 | − 1.34·2-s + 2.05·3-s − 0.190·4-s − 3.56·5-s − 2.75·6-s + 2.94·8-s + 1.20·9-s + 4.79·10-s − 1.27·11-s − 0.390·12-s − 7.31·15-s − 3.58·16-s + 7.73·17-s − 1.61·18-s + 0.943·19-s + 0.679·20-s + 1.72·22-s + 1.64·23-s + 6.04·24-s + 7.72·25-s − 3.68·27-s + 4.04·29-s + 9.83·30-s + 5.15·31-s − 1.07·32-s − 2.62·33-s − 10.4·34-s + ⋯ |
L(s) = 1 | − 0.951·2-s + 1.18·3-s − 0.0951·4-s − 1.59·5-s − 1.12·6-s + 1.04·8-s + 0.400·9-s + 1.51·10-s − 0.385·11-s − 0.112·12-s − 1.88·15-s − 0.895·16-s + 1.87·17-s − 0.381·18-s + 0.216·19-s + 0.151·20-s + 0.366·22-s + 0.343·23-s + 1.23·24-s + 1.54·25-s − 0.709·27-s + 0.751·29-s + 1.79·30-s + 0.925·31-s − 0.189·32-s − 0.456·33-s − 1.78·34-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
\(L(1)\) |
\(\approx\) |
\(1.052181028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.052181028\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 0.943T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 - 0.894T + 47T^{2} \) |
| 53 | \( 1 + 0.0799T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 7.62T + 61T^{2} \) |
| 67 | \( 1 + 6.32T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.760T + 73T^{2} \) |
| 79 | \( 1 + 2.85T + 79T^{2} \) |
| 83 | \( 1 - 2.32T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 - 0.478T + 97T^{2} \) |
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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
−7.82763003623370425339342388455, −7.71032149283301187886137669205, −6.95124442884721421073549768160, −5.71430870872087461492720318104, −4.73007036884230651743062489562, −4.16435157362806751551135311462, −3.25769740692404064816255469983, −2.92283130191855076660257286345, −1.51976802435098711799672306096, −0.58273335385365322963524140050,
0.58273335385365322963524140050, 1.51976802435098711799672306096, 2.92283130191855076660257286345, 3.25769740692404064816255469983, 4.16435157362806751551135311462, 4.73007036884230651743062489562, 5.71430870872087461492720318104, 6.95124442884721421073549768160, 7.71032149283301187886137669205, 7.82763003623370425339342388455