| L(s) = 1 | − 0.0812·2-s − 3-s − 1.99·4-s − 5-s + 0.0812·6-s + 2.03·7-s + 0.324·8-s + 9-s + 0.0812·10-s − 2.94·11-s + 1.99·12-s − 6.76·13-s − 0.165·14-s + 15-s + 3.96·16-s + 0.580·17-s − 0.0812·18-s − 5.44·19-s + 1.99·20-s − 2.03·21-s + 0.239·22-s − 6.31·23-s − 0.324·24-s + 25-s + 0.550·26-s − 27-s − 4.05·28-s + ⋯ |
| L(s) = 1 | − 0.0574·2-s − 0.577·3-s − 0.996·4-s − 0.447·5-s + 0.0331·6-s + 0.768·7-s + 0.114·8-s + 0.333·9-s + 0.0257·10-s − 0.889·11-s + 0.575·12-s − 1.87·13-s − 0.0441·14-s + 0.258·15-s + 0.990·16-s + 0.140·17-s − 0.0191·18-s − 1.24·19-s + 0.445·20-s − 0.443·21-s + 0.0511·22-s − 1.31·23-s − 0.0662·24-s + 0.200·25-s + 0.107·26-s − 0.192·27-s − 0.765·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2712548729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2712548729\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0812T + 2T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 6.76T + 13T^{2} \) |
| 17 | \( 1 - 0.580T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 29 | \( 1 + 0.882T + 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 + 4.00T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 6.23T + 67T^{2} \) |
| 71 | \( 1 - 7.99T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 - 5.03T + 97T^{2} \) |
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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
−7.901556270338651581788623624837, −7.69164764346302798214430372228, −6.71405888263965086494361829218, −5.66730344963324214873487219096, −5.14205133049825146890062482108, −4.49884860320251577253384688549, −4.01851234501557150784640180460, −2.72391501406888977142091143494, −1.77385218189770338473476042029, −0.27778644482252475086892375986,
0.27778644482252475086892375986, 1.77385218189770338473476042029, 2.72391501406888977142091143494, 4.01851234501557150784640180460, 4.49884860320251577253384688549, 5.14205133049825146890062482108, 5.66730344963324214873487219096, 6.71405888263965086494361829218, 7.69164764346302798214430372228, 7.901556270338651581788623624837
