| L(s) = 1 | + 5.19·7-s − 18.9i·11-s − 13i·13-s + 32.8i·17-s + 15.5i·19-s − 18.9·23-s + 32.8·29-s − 32.9i·31-s − 46i·37-s + 32.8·41-s − 12.1·43-s + 37.9·47-s − 22·49-s − 32.8i·53-s − 37.9i·59-s + ⋯ |
| L(s) = 1 | + 0.742·7-s − 1.72i·11-s − i·13-s + 1.93i·17-s + 0.820i·19-s − 0.824·23-s + 1.13·29-s − 1.06i·31-s − 1.24i·37-s + 0.801·41-s − 0.281·43-s + 0.807·47-s − 0.448·49-s − 0.620i·53-s − 0.643i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.918735786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918735786\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.19T + 49T^{2} \) |
| 11 | \( 1 + 18.9iT - 121T^{2} \) |
| 13 | \( 1 + 13iT - 169T^{2} \) |
| 17 | \( 1 - 32.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.5iT - 361T^{2} \) |
| 23 | \( 1 + 18.9T + 529T^{2} \) |
| 29 | \( 1 - 32.8T + 841T^{2} \) |
| 31 | \( 1 + 32.9iT - 961T^{2} \) |
| 37 | \( 1 + 46iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 12.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 26iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 131.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 23iT - 9.40e3T^{2} \) |
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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
−8.169827819146310001318453640673, −7.84413422629064177152039725242, −6.50608227025537354150694688200, −5.81350229196086384846136013132, −5.46145110872083071298144444948, −4.11804916511483357643799617444, −3.64403057570526523920833057234, −2.54888482309687695453161670349, −1.46931292776319012223553591469, −0.44169261506438937017184358995,
1.09771286566778736944942433421, 2.11532542538214519356110871240, 2.83460488431978798219136017002, 4.27965840195847051551827265284, 4.69655446787212737171889653892, 5.30036334452599124725885464168, 6.71784831197975959285634027989, 6.95325693825758336243324956452, 7.76514661264565286661991379273, 8.560448478634904832275315976128
