| L(s) = 1 | − 2-s − 0.280·3-s + 4-s + 0.181·5-s + 0.280·6-s − 8-s − 2.92·9-s − 0.181·10-s − 3.12·11-s − 0.280·12-s + 1.87·13-s − 0.0511·15-s + 16-s + 3.78·17-s + 2.92·18-s − 0.436·19-s + 0.181·20-s + 3.12·22-s − 1.37·23-s + 0.280·24-s − 4.96·25-s − 1.87·26-s + 1.66·27-s + 4.90·29-s + 0.0511·30-s + 5.31·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.162·3-s + 0.5·4-s + 0.0813·5-s + 0.114·6-s − 0.353·8-s − 0.973·9-s − 0.0575·10-s − 0.941·11-s − 0.0811·12-s + 0.518·13-s − 0.0132·15-s + 0.250·16-s + 0.916·17-s + 0.688·18-s − 0.100·19-s + 0.0406·20-s + 0.665·22-s − 0.286·23-s + 0.0573·24-s − 0.993·25-s − 0.366·26-s + 0.320·27-s + 0.910·29-s + 0.00933·30-s + 0.954·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.280T + 3T^{2} \) |
| 5 | \( 1 - 0.181T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 + 0.436T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 + 0.923T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 + 0.417T + 73T^{2} \) |
| 79 | \( 1 + 9.33T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 2.15T + 97T^{2} \) |
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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
−8.148214832032631535331729211745, −7.62318990712624990890177505543, −6.57817709069038039633573359779, −5.89207266335517286164860303895, −5.35531340308070544442622831668, −4.25321703931690148080972883784, −3.10384381741763834842909727750, −2.50593067165893513346337076491, −1.23133926828967034758837369092, 0,
1.23133926828967034758837369092, 2.50593067165893513346337076491, 3.10384381741763834842909727750, 4.25321703931690148080972883784, 5.35531340308070544442622831668, 5.89207266335517286164860303895, 6.57817709069038039633573359779, 7.62318990712624990890177505543, 8.148214832032631535331729211745
