| L(s) = 1 | − 0.776·2-s − 1.79·3-s − 1.39·4-s − 0.637·5-s + 1.39·6-s − 1.38·7-s + 2.63·8-s + 0.225·9-s + 0.495·10-s − 0.172·11-s + 2.50·12-s + 2.97·13-s + 1.07·14-s + 1.14·15-s + 0.745·16-s + 8.03·17-s − 0.175·18-s − 1.65·19-s + 0.890·20-s + 2.48·21-s + 0.133·22-s − 0.642·23-s − 4.73·24-s − 4.59·25-s − 2.30·26-s + 4.98·27-s + 1.93·28-s + ⋯ |
| L(s) = 1 | − 0.549·2-s − 1.03·3-s − 0.698·4-s − 0.285·5-s + 0.569·6-s − 0.522·7-s + 0.932·8-s + 0.0752·9-s + 0.156·10-s − 0.0519·11-s + 0.724·12-s + 0.823·13-s + 0.286·14-s + 0.295·15-s + 0.186·16-s + 1.94·17-s − 0.0412·18-s − 0.380·19-s + 0.199·20-s + 0.541·21-s + 0.0285·22-s − 0.133·23-s − 0.967·24-s − 0.918·25-s − 0.452·26-s + 0.958·27-s + 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4862155077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4862155077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 \) |
| good | 2 | \( 1 + 0.776T + 2T^{2} \) |
| 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 + 0.637T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 0.172T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 - 8.03T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 0.642T + 23T^{2} \) |
| 29 | \( 1 + 0.820T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.55T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 + 6.51T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 2.25T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.383465701905733764968003680380, −8.004403443390242870757512048837, −7.06534699568884962984767858807, −6.16898276966413424640612797755, −5.58116726431038102468179887894, −4.89385847517992552814869253179, −3.88427478975501053879825663969, −3.20882422040848766568406616506, −1.49932887815193903096023702308, −0.49998389764374305318866992777,
0.49998389764374305318866992777, 1.49932887815193903096023702308, 3.20882422040848766568406616506, 3.88427478975501053879825663969, 4.89385847517992552814869253179, 5.58116726431038102468179887894, 6.16898276966413424640612797755, 7.06534699568884962984767858807, 8.004403443390242870757512048837, 8.383465701905733764968003680380
