| L(s) = 1 | + 2-s + 2.71·3-s + 4-s − 0.231·5-s + 2.71·6-s − 3.53·7-s + 8-s + 4.36·9-s − 0.231·10-s − 4.96·11-s + 2.71·12-s − 3.53·14-s − 0.628·15-s + 16-s + 0.380·17-s + 4.36·18-s + 19-s − 0.231·20-s − 9.59·21-s − 4.96·22-s − 8.03·23-s + 2.71·24-s − 4.94·25-s + 3.71·27-s − 3.53·28-s + 3.83·29-s − 0.628·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.56·3-s + 0.5·4-s − 0.103·5-s + 1.10·6-s − 1.33·7-s + 0.353·8-s + 1.45·9-s − 0.0732·10-s − 1.49·11-s + 0.783·12-s − 0.944·14-s − 0.162·15-s + 0.250·16-s + 0.0923·17-s + 1.02·18-s + 0.229·19-s − 0.0517·20-s − 2.09·21-s − 1.05·22-s − 1.67·23-s + 0.554·24-s − 0.989·25-s + 0.714·27-s − 0.668·28-s + 0.712·29-s − 0.114·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.231T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 17 | \( 1 - 0.380T + 17T^{2} \) |
| 23 | \( 1 + 8.03T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 + 9.50T + 31T^{2} \) |
| 37 | \( 1 - 0.944T + 37T^{2} \) |
| 41 | \( 1 - 8.53T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 0.0285T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 1.08T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 0.0332T + 79T^{2} \) |
| 83 | \( 1 + 0.0511T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 2.09T + 97T^{2} \) |
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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
−7.64151465052453477959350876983, −7.23058211189805118084098085708, −6.08493306945525209545546997442, −5.69260514105686107214429641944, −4.49332411030653973819107201949, −3.80689726287162339047010778084, −3.15711550755016728424497297546, −2.60895393647077067326608526099, −1.87558575456221509625589517261, 0,
1.87558575456221509625589517261, 2.60895393647077067326608526099, 3.15711550755016728424497297546, 3.80689726287162339047010778084, 4.49332411030653973819107201949, 5.69260514105686107214429641944, 6.08493306945525209545546997442, 7.23058211189805118084098085708, 7.64151465052453477959350876983
