| L(s) = 1 | − 0.732·3-s + 0.386·5-s + 1.44·7-s − 2.46·9-s + 5.78·11-s + 4.84·13-s − 0.282·15-s + 3.82·17-s + 1.33·19-s − 1.05·21-s − 7.29·23-s − 4.85·25-s + 4·27-s − 3.50·29-s − 31-s − 4.23·33-s + 0.557·35-s − 6.02·37-s − 3.54·39-s + 4.49·41-s + 9.50·43-s − 0.952·45-s + 11.8·47-s − 4.91·49-s − 2.80·51-s − 12.8·53-s + 2.23·55-s + ⋯ |
| L(s) = 1 | − 0.422·3-s + 0.172·5-s + 0.545·7-s − 0.821·9-s + 1.74·11-s + 1.34·13-s − 0.0730·15-s + 0.928·17-s + 0.305·19-s − 0.230·21-s − 1.52·23-s − 0.970·25-s + 0.769·27-s − 0.650·29-s − 0.179·31-s − 0.737·33-s + 0.0942·35-s − 0.990·37-s − 0.567·39-s + 0.702·41-s + 1.44·43-s − 0.141·45-s + 1.72·47-s − 0.702·49-s − 0.392·51-s − 1.76·53-s + 0.301·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.838923372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838923372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 - 0.386T + 5T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 5.78T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 1.33T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 37 | \( 1 + 6.02T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 - 4.84T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 0.490T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−9.186421336169772326283071490871, −8.385184797339336926742484468600, −7.73113849346652108854676592003, −6.57081127142008431815421879374, −5.94984048196945630906118568500, −5.41332185046251654202503580637, −4.04176653233912767859737101029, −3.58251358514031890708405517189, −2.01345407707937718824743441463, −0.989807352561396833109664729646,
0.989807352561396833109664729646, 2.01345407707937718824743441463, 3.58251358514031890708405517189, 4.04176653233912767859737101029, 5.41332185046251654202503580637, 5.94984048196945630906118568500, 6.57081127142008431815421879374, 7.73113849346652108854676592003, 8.385184797339336926742484468600, 9.186421336169772326283071490871
