| L(s) = 1 | − 7-s − i·11-s + (1 − i)17-s + (1 − i)19-s + (1 − i)23-s + i·25-s + (1 + i)29-s − i·37-s + i·41-s − 47-s − 53-s − 71-s − i·73-s + i·77-s + (−1 + i)79-s + ⋯ |
| L(s) = 1 | − 7-s − i·11-s + (1 − i)17-s + (1 − i)19-s + (1 − i)23-s + i·25-s + (1 + i)29-s − i·37-s + i·41-s − 47-s − 53-s − 71-s − i·73-s + i·77-s + (−1 + i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9901232450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9901232450\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (1 - i)T - iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (-1 - i)T + iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.495176796307526365935178089189, −9.170113604138990475444606838205, −8.133082234759844983683208198924, −7.15344682254825108313963550289, −6.54872252403708265350922844956, −5.51871430423461080657967806116, −4.78343392773693847815040386442, −3.17552760762104351355673672730, −3.04043126554928920221463277642, −0.948801273017055576328193321162,
1.49503390060530460768852581895, 2.94629219843390123621995246818, 3.74602960213799912227260790124, 4.85476460581243332753220651953, 5.86726542001514331009570654582, 6.57289365047428225011075757892, 7.51902281514363926531244768445, 8.184676394905054721283816671312, 9.298426370763848530811450167283, 10.03004208454528131236286528271
