L(s) = 1 | + (−0.707 + 0.707i)2-s + i·3-s − 1.00i·4-s + (−0.461 − 0.461i)5-s + (−0.707 − 0.707i)6-s + (−0.292 − 2.62i)7-s + (0.707 + 0.707i)8-s − 9-s + 0.652·10-s + (1.60 + 1.60i)11-s + 1.00·12-s + (−2.51 − 2.58i)13-s + (2.06 + 1.65i)14-s + (0.461 − 0.461i)15-s − 1.00·16-s − 6.40·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + 0.577i·3-s − 0.500i·4-s + (−0.206 − 0.206i)5-s + (−0.288 − 0.288i)6-s + (−0.110 − 0.993i)7-s + (0.250 + 0.250i)8-s − 0.333·9-s + 0.206·10-s + (0.483 + 0.483i)11-s + 0.288·12-s + (−0.697 − 0.716i)13-s + (0.552 + 0.441i)14-s + (0.119 − 0.119i)15-s − 0.250·16-s − 1.55·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.544582 - 0.364192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.544582 - 0.364192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.292 + 2.62i)T \) |
| 13 | \( 1 + (2.51 + 2.58i)T \) |
good | 5 | \( 1 + (0.461 + 0.461i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.60 - 1.60i)T + 11iT^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 + (2.52 + 2.52i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 7.91T + 29T^{2} \) |
| 31 | \( 1 + (0.922 + 0.922i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.953 + 0.953i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.89 - 3.89i)T + 41iT^{2} \) |
| 43 | \( 1 - 4.17iT - 43T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 59 | \( 1 + (1.17 - 1.17i)T - 59iT^{2} \) |
| 61 | \( 1 + 14.4iT - 61T^{2} \) |
| 67 | \( 1 + (-3.47 + 3.47i)T - 67iT^{2} \) |
| 71 | \( 1 + (8.29 - 8.29i)T - 71iT^{2} \) |
| 73 | \( 1 + (-5.89 + 5.89i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.18T + 79T^{2} \) |
| 83 | \( 1 + (9.98 + 9.98i)T + 83iT^{2} \) |
| 89 | \( 1 + (8.54 - 8.54i)T - 89iT^{2} \) |
| 97 | \( 1 + (0.272 + 0.272i)T + 97iT^{2} \) |
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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
−10.46730681377561291714448461244, −9.815190958405368704859521062433, −8.793241790130626806207708718664, −8.109996113847413476731571953226, −6.93493139478037860224848444231, −6.36004998438604522047091437824, −4.65991497440114923613418241830, −4.37948263594754225822467239636, −2.55922774945430725825714847879, −0.43522153291359781130455939766,
1.73145345067835420253749231736, 2.77303285764392596223800217117, 4.07670298099729374818630133008, 5.52700044943962693844090223913, 6.62333993375499515977084849288, 7.38241457287301923490054509482, 8.636141410495913715976471791389, 8.988585283372242505265933211608, 10.03892395401412251364982348823, 11.24023554757454296538690732027