| L(s) = 1 | − 2.95·3-s + 20.3·5-s + 22.2·7-s − 18.2·9-s + 41.8·11-s + 35.3·13-s − 59.9·15-s − 17·17-s + 67.1·19-s − 65.7·21-s − 64.6·23-s + 287.·25-s + 133.·27-s + 83.0·29-s + 239.·31-s − 123.·33-s + 452.·35-s + 49.0·37-s − 104.·39-s − 225.·41-s + 83.4·43-s − 371.·45-s − 54.9·47-s + 153.·49-s + 50.1·51-s − 641.·53-s + 850.·55-s + ⋯ |
| L(s) = 1 | − 0.567·3-s + 1.81·5-s + 1.20·7-s − 0.677·9-s + 1.14·11-s + 0.753·13-s − 1.03·15-s − 0.242·17-s + 0.810·19-s − 0.683·21-s − 0.585·23-s + 2.30·25-s + 0.952·27-s + 0.531·29-s + 1.38·31-s − 0.651·33-s + 2.18·35-s + 0.217·37-s − 0.428·39-s − 0.860·41-s + 0.295·43-s − 1.23·45-s − 0.170·47-s + 0.448·49-s + 0.137·51-s − 1.66·53-s + 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.274956436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.274956436\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 2.95T + 27T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 - 41.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 67.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 225.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 83.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 641.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 727.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 868.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 640.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 19.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 619.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−9.478408533694521351122579744504, −8.802261065250953256398746210382, −7.982254025097485328240210322756, −6.47849573828911162371111200123, −6.20276230661461461662593824194, −5.28694386814790069192733950499, −4.57834163481095131749369309968, −3.01243779693290397548034328387, −1.77647500851079694127639528585, −1.10166980664016562389211910536,
1.10166980664016562389211910536, 1.77647500851079694127639528585, 3.01243779693290397548034328387, 4.57834163481095131749369309968, 5.28694386814790069192733950499, 6.20276230661461461662593824194, 6.47849573828911162371111200123, 7.982254025097485328240210322756, 8.802261065250953256398746210382, 9.478408533694521351122579744504
