| L(s) = 1 | + 2-s + 0.694·3-s + 4-s − 1.16·5-s + 0.694·6-s − 3.54·7-s + 8-s − 2.51·9-s − 1.16·10-s − 1.78·11-s + 0.694·12-s + 1.49·13-s − 3.54·14-s − 0.810·15-s + 16-s + 6.46·17-s − 2.51·18-s − 3.17·19-s − 1.16·20-s − 2.45·21-s − 1.78·22-s − 23-s + 0.694·24-s − 3.63·25-s + 1.49·26-s − 3.83·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.400·3-s + 0.5·4-s − 0.522·5-s + 0.283·6-s − 1.33·7-s + 0.353·8-s − 0.839·9-s − 0.369·10-s − 0.538·11-s + 0.200·12-s + 0.414·13-s − 0.946·14-s − 0.209·15-s + 0.250·16-s + 1.56·17-s − 0.593·18-s − 0.728·19-s − 0.261·20-s − 0.536·21-s − 0.380·22-s − 0.208·23-s + 0.141·24-s − 0.727·25-s + 0.293·26-s − 0.737·27-s − 0.669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183668347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183668347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.694T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 29 | \( 1 + 0.878T + 29T^{2} \) |
| 31 | \( 1 - 0.0973T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 7.77T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.88820831212904647087212767304, −7.45100071223206747309160307989, −6.54377350362563556077287735314, −5.73228522343997930894369778673, −5.51406587478572108341117964288, −4.07695625911881715264871407174, −3.70045872675573746280005436123, −2.92687285349347021716548394980, −2.29138834149547385325787131968, −0.64967371110841455582809543019,
0.64967371110841455582809543019, 2.29138834149547385325787131968, 2.92687285349347021716548394980, 3.70045872675573746280005436123, 4.07695625911881715264871407174, 5.51406587478572108341117964288, 5.73228522343997930894369778673, 6.54377350362563556077287735314, 7.45100071223206747309160307989, 7.88820831212904647087212767304
