| L(s) = 1 | − 2.57·2-s − 0.743·3-s + 4.64·4-s − 3.77·5-s + 1.91·6-s − 6.81·8-s − 2.44·9-s + 9.73·10-s − 5.38·11-s − 3.45·12-s + 2.81·15-s + 8.28·16-s − 6.31·17-s + 6.30·18-s − 3.32·19-s − 17.5·20-s + 13.8·22-s − 1.01·23-s + 5.06·24-s + 9.27·25-s + 4.05·27-s + 0.452·29-s − 7.24·30-s − 0.481·31-s − 7.71·32-s + 4.00·33-s + 16.2·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.429·3-s + 2.32·4-s − 1.68·5-s + 0.782·6-s − 2.40·8-s − 0.815·9-s + 3.07·10-s − 1.62·11-s − 0.997·12-s + 0.725·15-s + 2.07·16-s − 1.53·17-s + 1.48·18-s − 0.762·19-s − 3.92·20-s + 2.96·22-s − 0.210·23-s + 1.03·24-s + 1.85·25-s + 0.779·27-s + 0.0839·29-s − 1.32·30-s − 0.0865·31-s − 1.36·32-s + 0.697·33-s + 2.79·34-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 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.743T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 0.452T + 29T^{2} \) |
| 31 | \( 1 + 0.481T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 0.112T + 41T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 - 0.704T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.284T + 97T^{2} \) |
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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
−7.73902762646167504352075079820, −6.96160999794630242954622976727, −6.58482299074586776420474535256, −5.49341623262561385323365955109, −4.72432039542422974456455689691, −3.68986987837517059849897419774, −2.75119469139354507523898999453, −2.10575941716574087849358999105, −0.53264762089338003228513204836, 0,
0.53264762089338003228513204836, 2.10575941716574087849358999105, 2.75119469139354507523898999453, 3.68986987837517059849897419774, 4.72432039542422974456455689691, 5.49341623262561385323365955109, 6.58482299074586776420474535256, 6.96160999794630242954622976727, 7.73902762646167504352075079820
