| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4.23·7-s − 8-s − 2·9-s − 10-s + 3·11-s + 12-s + 0.236·13-s + 4.23·14-s + 15-s + 16-s − 1.14·17-s + 2·18-s − 6.47·19-s + 20-s − 4.23·21-s − 3·22-s + 1.14·23-s − 24-s + 25-s − 0.236·26-s − 5·27-s − 4.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.60·7-s − 0.353·8-s − 0.666·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.0654·13-s + 1.13·14-s + 0.258·15-s + 0.250·16-s − 0.277·17-s + 0.471·18-s − 1.48·19-s + 0.223·20-s − 0.924·21-s − 0.639·22-s + 0.238·23-s − 0.204·24-s + 0.200·25-s − 0.0462·26-s − 0.962·27-s − 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9931225730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9931225730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 37 | \( 1 + 7.94T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 1.85T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4.38T + 79T^{2} \) |
| 83 | \( 1 - 0.708T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
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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
−7.78691393395075565987888678837, −6.92160161182102774993918191113, −6.30642692930879235590851674865, −6.12498319989366140345176256922, −5.00234957259385455347198496386, −3.80177707837177269023637324294, −3.37655978446979920846907163676, −2.49617496702618766571630428197, −1.85651769414923188852093045253, −0.48555217773495951058936874804,
0.48555217773495951058936874804, 1.85651769414923188852093045253, 2.49617496702618766571630428197, 3.37655978446979920846907163676, 3.80177707837177269023637324294, 5.00234957259385455347198496386, 6.12498319989366140345176256922, 6.30642692930879235590851674865, 6.92160161182102774993918191113, 7.78691393395075565987888678837
