| L(s) = 1 | + 2.11·2-s + 3-s + 2.47·4-s − 0.285·5-s + 2.11·6-s + 0.998·8-s + 9-s − 0.602·10-s − 3.38·11-s + 2.47·12-s + 0.720·13-s − 0.285·15-s − 2.83·16-s − 4.26·17-s + 2.11·18-s − 6.19·19-s − 0.704·20-s − 7.16·22-s − 3.13·23-s + 0.998·24-s − 4.91·25-s + 1.52·26-s + 27-s + 0.696·29-s − 0.602·30-s + 9.88·31-s − 7.98·32-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.23·4-s − 0.127·5-s + 0.863·6-s + 0.352·8-s + 0.333·9-s − 0.190·10-s − 1.02·11-s + 0.713·12-s + 0.199·13-s − 0.0735·15-s − 0.708·16-s − 1.03·17-s + 0.498·18-s − 1.42·19-s − 0.157·20-s − 1.52·22-s − 0.653·23-s + 0.203·24-s − 0.983·25-s + 0.298·26-s + 0.192·27-s + 0.129·29-s − 0.110·30-s + 1.77·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 + 0.285T + 5T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 0.720T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 0.696T + 29T^{2} \) |
| 31 | \( 1 - 9.88T + 31T^{2} \) |
| 37 | \( 1 - 0.244T + 37T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 + 8.72T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.74485495681790832934613945823, −6.60738665194966190141965506545, −6.35391750217875750645821807337, −5.42178960227201737978356507518, −4.57551185412073986828501776652, −4.22437845943882103421306745642, −3.32579397403601233495397620014, −2.54586363213468981231354880576, −1.94928147768142589503945219865, 0,
1.94928147768142589503945219865, 2.54586363213468981231354880576, 3.32579397403601233495397620014, 4.22437845943882103421306745642, 4.57551185412073986828501776652, 5.42178960227201737978356507518, 6.35391750217875750645821807337, 6.60738665194966190141965506545, 7.74485495681790832934613945823
