| L(s) = 1 | + 2·2-s + 4·4-s + 15.8·5-s + 8·8-s + 31.7·10-s − 57.3·11-s + 5.69·13-s + 16·16-s + 51.8·17-s + 16.2·19-s + 63.5·20-s − 114.·22-s + 213.·23-s + 127.·25-s + 11.3·26-s + 218.·29-s − 251.·31-s + 32·32-s + 103.·34-s + 386.·37-s + 32.4·38-s + 127.·40-s + 328.·41-s − 37.5·43-s − 229.·44-s + 426.·46-s + 254.·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.42·5-s + 0.353·8-s + 1.00·10-s − 1.57·11-s + 0.121·13-s + 0.250·16-s + 0.740·17-s + 0.195·19-s + 0.711·20-s − 1.11·22-s + 1.93·23-s + 1.02·25-s + 0.0859·26-s + 1.39·29-s − 1.45·31-s + 0.176·32-s + 0.523·34-s + 1.71·37-s + 0.138·38-s + 0.502·40-s + 1.25·41-s − 0.133·43-s − 0.786·44-s + 1.36·46-s + 0.791·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.397168466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.397168466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 + 57.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.69T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 836.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 343.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 341.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 865.T + 9.12e5T^{2} \) |
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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
−9.864783963039341036748943054560, −9.073222236486308508142557890791, −7.899962458693330902949627752818, −7.06001676019769261432222178661, −5.96101408151034563092602807112, −5.43890016548073156820113994508, −4.65211902230538311190657271746, −3.05861659661790534892130539246, −2.42743037877594988650576148684, −1.09765747474199421275310300103,
1.09765747474199421275310300103, 2.42743037877594988650576148684, 3.05861659661790534892130539246, 4.65211902230538311190657271746, 5.43890016548073156820113994508, 5.96101408151034563092602807112, 7.06001676019769261432222178661, 7.899962458693330902949627752818, 9.073222236486308508142557890791, 9.864783963039341036748943054560
