| L(s) = 1 | + 2-s + 0.628·3-s + 4-s + 1.15·5-s + 0.628·6-s + 3.36·7-s + 8-s − 2.60·9-s + 1.15·10-s + 4.12·11-s + 0.628·12-s + 4.23·13-s + 3.36·14-s + 0.726·15-s + 16-s + 3.54·17-s − 2.60·18-s + 19-s + 1.15·20-s + 2.11·21-s + 4.12·22-s + 8.51·23-s + 0.628·24-s − 3.66·25-s + 4.23·26-s − 3.52·27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.362·3-s + 0.5·4-s + 0.517·5-s + 0.256·6-s + 1.27·7-s + 0.353·8-s − 0.868·9-s + 0.365·10-s + 1.24·11-s + 0.181·12-s + 1.17·13-s + 0.900·14-s + 0.187·15-s + 0.250·16-s + 0.859·17-s − 0.613·18-s + 0.229·19-s + 0.258·20-s + 0.462·21-s + 0.879·22-s + 1.77·23-s + 0.128·24-s − 0.732·25-s + 0.831·26-s − 0.677·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.834877919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.834877919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.628T + 3T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 5.08T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 - 7.77T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.27T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 8.06T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 + 3.87T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−7.966445908862537626386616890601, −6.97535921014791220557053233530, −6.38618326777462871979309108367, −5.57239970232185622211315622357, −5.17270452845532435105095971311, −4.26042495111753351455652939430, −3.48016548601396999136170681841, −2.87057727454697446598362481009, −1.67292646562327670603483400708, −1.25109251324774523372979564217,
1.25109251324774523372979564217, 1.67292646562327670603483400708, 2.87057727454697446598362481009, 3.48016548601396999136170681841, 4.26042495111753351455652939430, 5.17270452845532435105095971311, 5.57239970232185622211315622357, 6.38618326777462871979309108367, 6.97535921014791220557053233530, 7.966445908862537626386616890601
