L(s) = 1 | + 4·5-s + 7-s + 14·11-s − 16.1i·13-s − 23·17-s + (−10 + 16.1i)19-s − 23-s − 9·25-s − 48.4i·29-s − 32.3i·31-s + 4·35-s − 32.3i·37-s + 32.3i·41-s − 68·43-s + 26·47-s + ⋯ |
L(s) = 1 | + 0.800·5-s + 0.142·7-s + 1.27·11-s − 1.24i·13-s − 1.35·17-s + (−0.526 + 0.850i)19-s − 0.0434·23-s − 0.359·25-s − 1.67i·29-s − 1.04i·31-s + 0.114·35-s − 0.873i·37-s + 0.788i·41-s − 1.58·43-s + 0.553·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.510767588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.510767588\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (10 - 16.1i)T \) |
good | 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - T + 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 + 16.1iT - 169T^{2} \) |
| 17 | \( 1 + 23T + 289T^{2} \) |
| 23 | \( 1 + T + 529T^{2} \) |
| 29 | \( 1 + 48.4iT - 841T^{2} \) |
| 31 | \( 1 + 32.3iT - 961T^{2} \) |
| 37 | \( 1 + 32.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68T + 1.84e3T^{2} \) |
| 47 | \( 1 - 26T + 2.20e3T^{2} \) |
| 53 | \( 1 + 80.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 32T + 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 96.9iT - 9.40e3T^{2} \) |
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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
−8.340608077545456062143917821829, −7.80263981184762920350889226268, −6.58560564364407808558447519741, −6.20184420731952870582747070259, −5.41738454088237330279949689512, −4.36591277665851507663040142204, −3.65400151638158908060884796036, −2.39601033317843004044736227317, −1.66059932331320012918791253669, −0.31635098721583418729892834707,
1.43261392104023903264379396732, 2.01974818398281076506408784461, 3.23408935616517252793460482226, 4.33051914334330993187910979470, 4.85569411652833833339777076578, 5.99832067404944156345883922116, 6.79921111500720999972393408852, 6.93712963739735176331479308540, 8.431785666303300060853522271220, 9.092292073977344619226420537078