| L(s) = 1 | + 1.28i·3-s − 1.58i·5-s − 1.86·7-s + 1.34·9-s − 0.322i·13-s + 2.03·15-s − 4.79·17-s − 3.38i·19-s − 2.40i·21-s − 3.08·23-s + 2.49·25-s + 5.58i·27-s + 10.4i·29-s + 2.82·31-s + 2.95i·35-s + ⋯ |
| L(s) = 1 | + 0.742i·3-s − 0.707i·5-s − 0.706·7-s + 0.448·9-s − 0.0893i·13-s + 0.525·15-s − 1.16·17-s − 0.776i·19-s − 0.524i·21-s − 0.642·23-s + 0.499·25-s + 1.07i·27-s + 1.93i·29-s + 0.506·31-s + 0.499i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642705650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642705650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.28iT - 3T^{2} \) |
| 5 | \( 1 + 1.58iT - 5T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 13 | \( 1 + 0.322iT - 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 + 3.38iT - 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 10.4iT - 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.57iT - 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 + 3.61iT - 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 5.38iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 5.97T + 79T^{2} \) |
| 83 | \( 1 + 0.139iT - 83T^{2} \) |
| 89 | \( 1 - 4.28T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.845801028306270590998803315897, −7.88091969288846463184995949223, −6.87810880287321255395343170959, −6.46313145948204869981284991432, −5.22761832873060003272979485431, −4.80300753719900881597025401565, −3.99213055519798057885344834664, −3.21135097150157044282960825818, −2.07971319911654571438348743164, −0.75285301824138727534315666219,
0.71150691335577342515415993470, 2.08866389139288901295802860397, 2.66406702943345438643949575406, 3.86506457504027702323870031237, 4.42685711645595034996344435904, 5.82636942114321960520203680230, 6.33545320982591088815334583596, 6.91195492894231923358089011517, 7.60045426301554049763447459207, 8.237292119557547954565521323114
