L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (2.27 + 2.27i)5-s + (−2.27 + 1.35i)7-s + (0.707 + 0.707i)8-s − 3.21·10-s + (−0.355 − 0.355i)11-s + (−2 + 3i)13-s + (0.648 − 2.56i)14-s − 1.00·16-s − 4.32·17-s + (−5.98 − 5.98i)19-s + (2.27 − 2.27i)20-s + 0.502·22-s − 2.38i·23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.01 + 1.01i)5-s + (−0.858 + 0.512i)7-s + (0.250 + 0.250i)8-s − 1.01·10-s + (−0.107 − 0.107i)11-s + (−0.554 + 0.832i)13-s + (0.173 − 0.685i)14-s − 0.250·16-s − 1.04·17-s + (−1.37 − 1.37i)19-s + (0.508 − 0.508i)20-s + 0.107·22-s − 0.497i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1173680041\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1173680041\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.27 - 1.35i)T \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 5 | \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.355 + 0.355i)T + 11iT^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + (5.98 + 5.98i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.38iT - 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + (1.08 + 1.08i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.18 + 5.18i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.53 - 3.53i)T + 41iT^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 + (4.71 - 4.71i)T - 47iT^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + (-3.61 + 3.61i)T - 59iT^{2} \) |
| 61 | \( 1 - 4.32iT - 61T^{2} \) |
| 67 | \( 1 + (0.531 - 0.531i)T - 67iT^{2} \) |
| 71 | \( 1 + (6.38 - 6.38i)T - 71iT^{2} \) |
| 73 | \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + (-6.71 - 6.71i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.75 - 3.75i)T - 89iT^{2} \) |
| 97 | \( 1 + (-12.9 - 12.9i)T + 97iT^{2} \) |
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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
−9.734016290919285655915402174246, −9.118343004010098623307957572857, −8.602782699256789705950832681855, −7.14271523171120845253814182947, −6.71935713868429232870237368413, −6.20736103915339714107801596930, −5.26734373415024621704188902386, −4.11390879097279039894249877800, −2.55649983421671664597647526787, −2.21285629414637577421651074708,
0.04907635350887480768583320430, 1.47042325372393017459749116107, 2.43165823447890582005879511761, 3.65037815432242970697011326578, 4.61110780356238839602296709001, 5.61416413831122771611788710745, 6.40113591371110159589424671408, 7.35523372685371351178164170250, 8.371415180994314999622231710018, 8.920734818804050147266982901970