| L(s) = 1 | + 1.78·2-s + 0.303·3-s + 1.20·4-s − 3.29·5-s + 0.542·6-s − 1.30·7-s − 1.42·8-s − 2.90·9-s − 5.90·10-s − 11-s + 0.365·12-s − 0.0355·13-s − 2.33·14-s − 15-s − 4.95·16-s + 1.92·17-s − 5.20·18-s − 2.26·19-s − 3.96·20-s − 0.395·21-s − 1.78·22-s + 2.13·23-s − 0.432·24-s + 5.87·25-s − 0.0636·26-s − 1.79·27-s − 1.56·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.175·3-s + 0.601·4-s − 1.47·5-s + 0.221·6-s − 0.492·7-s − 0.503·8-s − 0.969·9-s − 1.86·10-s − 0.301·11-s + 0.105·12-s − 0.00986·13-s − 0.623·14-s − 0.258·15-s − 1.23·16-s + 0.467·17-s − 1.22·18-s − 0.519·19-s − 0.887·20-s − 0.0862·21-s − 0.381·22-s + 0.445·23-s − 0.0882·24-s + 1.17·25-s − 0.0124·26-s − 0.344·27-s − 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 - 0.303T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 + 0.0355T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 - 0.728T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 - 4.22T + 59T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 9.50T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 0.802T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−10.29495042283018189904132731488, −8.971635020341445959958302314280, −8.317179474695420391534735611459, −7.31934675187941602863251432787, −6.31137767634865095583292220341, −5.33484025054038845679638157171, −4.37257687706191603074494445033, −3.48277964736747044957505906713, −2.81246843783612844611606967097, 0,
2.81246843783612844611606967097, 3.48277964736747044957505906713, 4.37257687706191603074494445033, 5.33484025054038845679638157171, 6.31137767634865095583292220341, 7.31934675187941602863251432787, 8.317179474695420391534735611459, 8.971635020341445959958302314280, 10.29495042283018189904132731488
