| L(s) = 1 | + 27.3·3-s + 52.2·5-s + 49·7-s + 503.·9-s + 142.·11-s + 219.·13-s + 1.42e3·15-s − 1.55e3·17-s + 372.·19-s + 1.33e3·21-s + 3.65e3·23-s − 397.·25-s + 7.11e3·27-s + 4.49e3·29-s − 9.15e3·31-s + 3.88e3·33-s + 2.55e3·35-s + 3.66e3·37-s + 5.99e3·39-s − 1.76e4·41-s + 1.10e4·43-s + 2.62e4·45-s − 1.97e4·47-s + 2.40e3·49-s − 4.25e4·51-s + 2.29e4·53-s + 7.43e3·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 0.934·5-s + 0.377·7-s + 2.07·9-s + 0.354·11-s + 0.359·13-s + 1.63·15-s − 1.30·17-s + 0.236·19-s + 0.662·21-s + 1.44·23-s − 0.127·25-s + 1.87·27-s + 0.991·29-s − 1.71·31-s + 0.621·33-s + 0.353·35-s + 0.440·37-s + 0.630·39-s − 1.64·41-s + 0.914·43-s + 1.93·45-s − 1.30·47-s + 0.142·49-s − 2.28·51-s + 1.12·53-s + 0.331·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.863446475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.863446475\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 27.3T + 243T^{2} \) |
| 5 | \( 1 - 52.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 142.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 219.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.55e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 372.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−11.23056085904376587399643954725, −10.12448742193179944551088137308, −9.108792998683760856927701438695, −8.747681196563211493632659002442, −7.53045761958641495070014471957, −6.49641546615463814146994746639, −4.88277312560471376528809797792, −3.58415971493482620999968387622, −2.40700461312426068217490948509, −1.48634184489224277341840464107,
1.48634184489224277341840464107, 2.40700461312426068217490948509, 3.58415971493482620999968387622, 4.88277312560471376528809797792, 6.49641546615463814146994746639, 7.53045761958641495070014471957, 8.747681196563211493632659002442, 9.108792998683760856927701438695, 10.12448742193179944551088137308, 11.23056085904376587399643954725
