L(s) = 1 | + 2-s + 4-s − 2i·5-s + 8-s − 9-s − 2i·10-s + 16-s − i·17-s − 18-s − 2i·20-s − 3·25-s + 32-s − i·34-s − 36-s − 2i·40-s + 2i·41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 2i·5-s + 8-s − 9-s − 2i·10-s + 16-s − i·17-s − 18-s − 2i·20-s − 3·25-s + 32-s − i·34-s − 36-s − 2i·40-s + 2i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.214623940\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214623940\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + 2iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−8.430633300389028200049488011712, −8.048066470631121885324039611231, −7.04115919702666259433507616079, −5.99821582395933511915893744867, −5.42316652431438731144289533028, −4.84989405626378519022573792706, −4.19642928662406139554867024405, −3.15546472061134942580040113142, −2.10603401765731893028781930880, −0.932916404743028582333596457171,
2.06773450539787846026591481102, 2.69363529614261304227035350493, 3.50207615625483771515222682221, 4.05546430443564320995967653251, 5.47196184114791683729290094088, 5.95268570667967206761893156753, 6.63573805565276938472088701074, 7.26929135940430910527311972030, 7.932166059349246756991103152183, 8.910798471797272718739241120261