| L(s) = 1 | + 0.851·2-s + 0.661·3-s − 1.27·4-s − 3.44·5-s + 0.562·6-s − 2.78·8-s − 2.56·9-s − 2.92·10-s + 0.897·11-s − 0.843·12-s − 2.27·15-s + 0.178·16-s + 1.93·17-s − 2.18·18-s − 1.03·19-s + 4.38·20-s + 0.764·22-s + 5.65·23-s − 1.84·24-s + 6.84·25-s − 3.67·27-s − 1.83·29-s − 1.93·30-s + 9.13·31-s + 5.72·32-s + 0.593·33-s + 1.64·34-s + ⋯ |
| L(s) = 1 | + 0.601·2-s + 0.381·3-s − 0.637·4-s − 1.53·5-s + 0.229·6-s − 0.985·8-s − 0.854·9-s − 0.926·10-s + 0.270·11-s − 0.243·12-s − 0.587·15-s + 0.0445·16-s + 0.469·17-s − 0.514·18-s − 0.238·19-s + 0.981·20-s + 0.162·22-s + 1.17·23-s − 0.376·24-s + 1.36·25-s − 0.707·27-s − 0.340·29-s − 0.353·30-s + 1.64·31-s + 1.01·32-s + 0.103·33-s + 0.282·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.851T + 2T^{2} \) |
| 3 | \( 1 - 0.661T + 3T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 - 0.897T + 11T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 - 9.13T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 + 3.91T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 - 0.510T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 0.506T + 97T^{2} \) |
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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
−7.67874176296820782330949643006, −6.73191409940418291303578177876, −6.01034570040773348166167579912, −5.17356813385525448016453962906, −4.46997754453368242486486327784, −3.97877340599758085936235890512, −3.13302140137461983027133819927, −2.79458043333370123706056089487, −1.01628741294118905949050866221, 0,
1.01628741294118905949050866221, 2.79458043333370123706056089487, 3.13302140137461983027133819927, 3.97877340599758085936235890512, 4.46997754453368242486486327784, 5.17356813385525448016453962906, 6.01034570040773348166167579912, 6.73191409940418291303578177876, 7.67874176296820782330949643006
