| L(s) = 1 | − 1.81·2-s − 3-s + 1.28·4-s − 5-s + 1.81·6-s + 1.28·8-s + 9-s + 1.81·10-s + 11-s − 1.28·12-s − 5.10·13-s + 15-s − 4.91·16-s + 1.91·17-s − 1.81·18-s − 3.33·19-s − 1.28·20-s − 1.81·22-s − 2.28·23-s − 1.28·24-s + 25-s + 9.25·26-s − 27-s + 1.18·29-s − 1.81·30-s + 2.52·31-s + 6.33·32-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.644·4-s − 0.447·5-s + 0.740·6-s + 0.455·8-s + 0.333·9-s + 0.573·10-s + 0.301·11-s − 0.372·12-s − 1.41·13-s + 0.258·15-s − 1.22·16-s + 0.464·17-s − 0.427·18-s − 0.765·19-s − 0.288·20-s − 0.386·22-s − 0.477·23-s − 0.263·24-s + 0.200·25-s + 1.81·26-s − 0.192·27-s + 0.220·29-s − 0.331·30-s + 0.453·31-s + 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 0.289T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 0.578T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.372T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 0.0297T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.53411127963037626620473089796, −7.07668039480061036754846151448, −6.32025728395790887130775215931, −5.45559091494706120771356843942, −4.55011768124583944340826935968, −4.15020133831600927385547182314, −2.82741351161471433709606991663, −1.94158163988211928467527780916, −0.885381829147084393037760214484, 0,
0.885381829147084393037760214484, 1.94158163988211928467527780916, 2.82741351161471433709606991663, 4.15020133831600927385547182314, 4.55011768124583944340826935968, 5.45559091494706120771356843942, 6.32025728395790887130775215931, 7.07668039480061036754846151448, 7.53411127963037626620473089796
