| L(s) = 1 | + 2.23·2-s + 0.618·3-s + 3.00·4-s − 5-s + 1.38·6-s + 7-s + 2.23·8-s − 2.61·9-s − 2.23·10-s + 1.85·12-s − 4.61·13-s + 2.23·14-s − 0.618·15-s − 0.999·16-s − 5.61·17-s − 5.85·18-s + 3.23·19-s − 3.00·20-s + 0.618·21-s − 4.47·23-s + 1.38·24-s + 25-s − 10.3·26-s − 3.47·27-s + 3.00·28-s − 0.854·29-s − 1.38·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.356·3-s + 1.50·4-s − 0.447·5-s + 0.564·6-s + 0.377·7-s + 0.790·8-s − 0.872·9-s − 0.707·10-s + 0.535·12-s − 1.28·13-s + 0.597·14-s − 0.159·15-s − 0.249·16-s − 1.36·17-s − 1.37·18-s + 0.742·19-s − 0.670·20-s + 0.134·21-s − 0.932·23-s + 0.282·24-s + 0.200·25-s − 2.02·26-s − 0.668·27-s + 0.566·28-s − 0.158·29-s − 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 0.854T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 4.76T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.38T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.86995788280951749911567349659, −7.13947161232887866348855133697, −6.43206868596616074682704105938, −5.55670280602890024036052318473, −4.95039088217076630484222308665, −4.29780562648481389884533158451, −3.49912845228479082262868943128, −2.68034246250065049868440454068, −2.04858530132355269273641636384, 0,
2.04858530132355269273641636384, 2.68034246250065049868440454068, 3.49912845228479082262868943128, 4.29780562648481389884533158451, 4.95039088217076630484222308665, 5.55670280602890024036052318473, 6.43206868596616074682704105938, 7.13947161232887866348855133697, 7.86995788280951749911567349659
