| L(s) = 1 | − 4-s − i·7-s − 2i·13-s + 16-s + 19-s + i·28-s − 31-s − i·37-s + i·43-s + 2i·52-s − 61-s − 64-s + 2i·67-s + i·73-s − 76-s + ⋯ |
| L(s) = 1 | − 4-s − i·7-s − 2i·13-s + 16-s + 19-s + i·28-s − 31-s − i·37-s + i·43-s + 2i·52-s − 61-s − 64-s + 2i·67-s + i·73-s − 76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7248099457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7248099457\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - 2iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−10.37597687462054513137667883576, −9.831807487832954854800311824728, −8.845288129942645932477566539947, −7.86139484924557572318788804059, −7.35532613837584218097585801942, −5.82383502029821587873126107837, −5.11346732251850265724749373303, −3.98437199804989189250443829933, −3.11645252466251850056974162544, −0.903546389881432883975382087892,
1.84901785243619914988913026027, 3.39282298873846328586922159066, 4.49721641591086906274051710305, 5.32222385461243599161022573612, 6.32122651159402923306347575626, 7.42626025789146920812590421178, 8.531185975237468514862768687538, 9.217022821958668079393325244446, 9.615608100842875545926687207280, 10.88001516629769638639021366239
