| L(s) = 1 | − 1.28·2-s − 3-s − 0.341·4-s − 1.67·5-s + 1.28·6-s + 2.27·7-s + 3.01·8-s + 9-s + 2.15·10-s − 11-s + 0.341·12-s − 2.92·14-s + 1.67·15-s − 3.20·16-s − 0.562·17-s − 1.28·18-s + 1.79·19-s + 0.569·20-s − 2.27·21-s + 1.28·22-s + 8.24·23-s − 3.01·24-s − 2.20·25-s − 27-s − 0.775·28-s − 0.867·29-s − 2.15·30-s + ⋯ |
| L(s) = 1 | − 0.910·2-s − 0.577·3-s − 0.170·4-s − 0.747·5-s + 0.525·6-s + 0.859·7-s + 1.06·8-s + 0.333·9-s + 0.680·10-s − 0.301·11-s + 0.0984·12-s − 0.782·14-s + 0.431·15-s − 0.800·16-s − 0.136·17-s − 0.303·18-s + 0.411·19-s + 0.127·20-s − 0.496·21-s + 0.274·22-s + 1.71·23-s − 0.615·24-s − 0.441·25-s − 0.192·27-s − 0.146·28-s − 0.161·29-s − 0.392·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 17 | \( 1 + 0.562T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 + 0.867T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 0.592T + 41T^{2} \) |
| 43 | \( 1 - 3.33T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 8.16T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 + 6.88T + 97T^{2} \) |
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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
−7.898475831514515703027358031876, −7.30126985530081830990347632003, −6.65705139352768353209275140000, −5.34311482682590287726375557565, −5.01901859698137297741842338067, −4.20246736109043460649590996762, −3.35450316646097791940761658786, −1.94842676665245518997299791207, −1.04599523800656973751134569036, 0,
1.04599523800656973751134569036, 1.94842676665245518997299791207, 3.35450316646097791940761658786, 4.20246736109043460649590996762, 5.01901859698137297741842338067, 5.34311482682590287726375557565, 6.65705139352768353209275140000, 7.30126985530081830990347632003, 7.898475831514515703027358031876
