| L(s) = 1 | + 2.79·2-s − 1.12·3-s + 5.83·4-s − 3.14·6-s + 10.7·8-s − 1.74·9-s + 2.92·11-s − 6.54·12-s + 3.08·13-s + 18.3·16-s − 4.87·18-s + 8.20·22-s − 12.0·24-s + 8.64·26-s + 5.32·27-s + 29.9·32-s − 3.28·33-s − 10.1·36-s + 9.15·37-s − 3.46·39-s + 17.0·44-s − 20.6·48-s − 7·49-s + 18.0·52-s − 13.6·53-s + 14.8·54-s + 1.11·61-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.647·3-s + 2.91·4-s − 1.28·6-s + 3.79·8-s − 0.580·9-s + 0.883·11-s − 1.89·12-s + 0.856·13-s + 4.59·16-s − 1.14·18-s + 1.74·22-s − 2.45·24-s + 1.69·26-s + 1.02·27-s + 5.29·32-s − 0.572·33-s − 1.69·36-s + 1.50·37-s − 0.555·39-s + 2.57·44-s − 2.97·48-s − 49-s + 2.49·52-s − 1.87·53-s + 2.02·54-s + 0.143·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.578169917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.578169917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.15T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 1.11T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.41233094834616903373073993671, −6.49848432228201199322677556068, −6.19746973791770010445400440692, −5.75545382980732090797083129961, −4.83663580188621937779473520248, −4.43474411663834681004913158026, −3.51553302617958991809570929779, −3.04244193615383636262625188057, −1.99856400356378144510660058153, −1.08118150142183632667605422274,
1.08118150142183632667605422274, 1.99856400356378144510660058153, 3.04244193615383636262625188057, 3.51553302617958991809570929779, 4.43474411663834681004913158026, 4.83663580188621937779473520248, 5.75545382980732090797083129961, 6.19746973791770010445400440692, 6.49848432228201199322677556068, 7.41233094834616903373073993671
