| L(s) = 1 | − 2-s − 1.74·3-s + 4-s + 2.20·5-s + 1.74·6-s + 0.156·7-s − 8-s + 0.0472·9-s − 2.20·10-s + 0.652·11-s − 1.74·12-s − 6.50·13-s − 0.156·14-s − 3.84·15-s + 16-s − 0.464·17-s − 0.0472·18-s + 5.83·19-s + 2.20·20-s − 0.273·21-s − 0.652·22-s − 8.89·23-s + 1.74·24-s − 0.138·25-s + 6.50·26-s + 5.15·27-s + 0.156·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.986·5-s + 0.712·6-s + 0.0592·7-s − 0.353·8-s + 0.0157·9-s − 0.697·10-s + 0.196·11-s − 0.503·12-s − 1.80·13-s − 0.0418·14-s − 0.993·15-s + 0.250·16-s − 0.112·17-s − 0.0111·18-s + 1.33·19-s + 0.493·20-s − 0.0596·21-s − 0.139·22-s − 1.85·23-s + 0.356·24-s − 0.0277·25-s + 1.27·26-s + 0.991·27-s + 0.0296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 7 | \( 1 - 0.156T + 7T^{2} \) |
| 11 | \( 1 - 0.652T + 11T^{2} \) |
| 13 | \( 1 + 6.50T + 13T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 - 0.0633T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 - 0.948T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 0.890T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 0.438T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−9.881372789015223040305866055191, −9.207747246209854060277454102041, −7.973780224038943052407613768862, −7.16453963526854979932401062916, −6.16498129826795321633908221129, −5.58078163031941193791032685974, −4.65568393819433903031570293196, −2.85688414970249145040367873531, −1.69916984989903885174403769138, 0,
1.69916984989903885174403769138, 2.85688414970249145040367873531, 4.65568393819433903031570293196, 5.58078163031941193791032685974, 6.16498129826795321633908221129, 7.16453963526854979932401062916, 7.973780224038943052407613768862, 9.207747246209854060277454102041, 9.881372789015223040305866055191
