| L(s) = 1 | + 2.24·2-s + 3-s + 3.04·4-s − 1.69·5-s + 2.24·6-s − 7-s + 2.35·8-s + 9-s − 3.80·10-s + 5.29·11-s + 3.04·12-s − 2.24·14-s − 1.69·15-s − 0.801·16-s − 2.24·17-s + 2.24·18-s + 7.49·19-s − 5.15·20-s − 21-s + 11.8·22-s + 6.76·23-s + 2.35·24-s − 2.13·25-s + 27-s − 3.04·28-s + 7.56·29-s − 3.80·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.52·4-s − 0.756·5-s + 0.917·6-s − 0.377·7-s + 0.833·8-s + 0.333·9-s − 1.20·10-s + 1.59·11-s + 0.880·12-s − 0.600·14-s − 0.436·15-s − 0.200·16-s − 0.544·17-s + 0.529·18-s + 1.71·19-s − 1.15·20-s − 0.218·21-s + 2.53·22-s + 1.41·23-s + 0.481·24-s − 0.427·25-s + 0.192·27-s − 0.576·28-s + 1.40·29-s − 0.694·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.554243201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.554243201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 7.49T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 - 6.16T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 4.37T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 + 0.131T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 - 0.374T + 97T^{2} \) |
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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
−8.539701629541701612554164882285, −7.55587629096224390133031898728, −6.81207275164612027348309744468, −6.42965657205684246404445650176, −5.29592096540882493916350343696, −4.61920792380952953244898928711, −3.75029792038671145366280476113, −3.42642323348322111601242899590, −2.52533045892818794950095249894, −1.15052802232486661425707848468,
1.15052802232486661425707848468, 2.52533045892818794950095249894, 3.42642323348322111601242899590, 3.75029792038671145366280476113, 4.61920792380952953244898928711, 5.29592096540882493916350343696, 6.42965657205684246404445650176, 6.81207275164612027348309744468, 7.55587629096224390133031898728, 8.539701629541701612554164882285
