| L(s) = 1 | − 2-s + 4-s − 0.494i·5-s − i·7-s − 8-s + 0.494i·10-s + (−0.197 − 3.31i)11-s + 5.50i·13-s + i·14-s + 16-s + 2.35·17-s + 1.14i·19-s − 0.494i·20-s + (0.197 + 3.31i)22-s − 4.19i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.220i·5-s − 0.377i·7-s − 0.353·8-s + 0.156i·10-s + (−0.0596 − 0.998i)11-s + 1.52i·13-s + 0.267i·14-s + 0.250·16-s + 0.571·17-s + 0.263i·19-s − 0.110i·20-s + (0.0421 + 0.705i)22-s − 0.874i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135146725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135146725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.197 + 3.31i)T \) |
| good | 5 | \( 1 + 0.494iT - 5T^{2} \) |
| 13 | \( 1 - 5.50iT - 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 1.14iT - 19T^{2} \) |
| 23 | \( 1 + 4.19iT - 23T^{2} \) |
| 29 | \( 1 - 0.175T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 9.72iT - 43T^{2} \) |
| 47 | \( 1 + 4.06iT - 47T^{2} \) |
| 53 | \( 1 + 2.97iT - 53T^{2} \) |
| 59 | \( 1 + 2.26iT - 59T^{2} \) |
| 61 | \( 1 + 4.51iT - 61T^{2} \) |
| 67 | \( 1 - 8.46T + 67T^{2} \) |
| 71 | \( 1 + 3.20iT - 71T^{2} \) |
| 73 | \( 1 - 2.39iT - 73T^{2} \) |
| 79 | \( 1 - 1.55iT - 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 + 2.32iT - 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.359308378994070919412058386964, −8.663493003931282821543777501508, −8.065423468977411097895958033148, −6.98200651873567093776473957546, −6.44310542031259912612972807335, −5.36851899194436541215351358088, −4.27755127724774655702597936572, −3.25766961181864843452958982665, −1.97801949166251975892182150567, −0.70253445142295536984028254453,
1.11353532415490018697083257760, 2.50762330641449763836393200582, 3.31726308681938476250511943742, 4.76319229485349375123988836361, 5.62704012127600326455896116058, 6.53399077604462053410114224797, 7.52620157645345095949774576111, 7.959057843883323229463653645686, 8.947581671765880285154177901385, 9.757324142821259797762816996288
