| L(s) = 1 | − 2.70·2-s + 3-s + 5.30·4-s − 2.70·6-s − 8.93·8-s + 9-s + 2.82·11-s + 5.30·12-s + 4.47·13-s + 13.5·16-s − 4.23·17-s − 2.70·18-s + 2.10·19-s − 7.63·22-s + 6.61·23-s − 8.93·24-s − 12.1·26-s + 27-s + 4·29-s + 9.33·31-s − 18.7·32-s + 2.82·33-s + 11.4·34-s + 5.30·36-s − 4.20·37-s − 5.67·38-s + 4.47·39-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.65·4-s − 1.10·6-s − 3.15·8-s + 0.333·9-s + 0.851·11-s + 1.53·12-s + 1.24·13-s + 3.38·16-s − 1.02·17-s − 0.637·18-s + 0.481·19-s − 1.62·22-s + 1.37·23-s − 1.82·24-s − 2.37·26-s + 0.192·27-s + 0.742·29-s + 1.67·31-s − 3.30·32-s + 0.491·33-s + 1.96·34-s + 0.884·36-s − 0.691·37-s − 0.920·38-s + 0.717·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183995587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183995587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 9.33T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 2.37T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 + 7.78T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 3.30T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 2.65T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.687705919100923045334291746031, −8.101682073533532335144848227539, −7.25330156203644376458178015440, −6.58227560531583162193366744856, −6.14102386796058597364996562557, −4.66554663858442666284954550972, −3.43759908369699768680896824807, −2.71055993464176665232134085869, −1.60646624997846904020052753148, −0.890310623491680992123692546607,
0.890310623491680992123692546607, 1.60646624997846904020052753148, 2.71055993464176665232134085869, 3.43759908369699768680896824807, 4.66554663858442666284954550972, 6.14102386796058597364996562557, 6.58227560531583162193366744856, 7.25330156203644376458178015440, 8.101682073533532335144848227539, 8.687705919100923045334291746031
