| L(s) = 1 | + i·2-s − 4-s − i·5-s − i·7-s − i·8-s + 10-s − i·11-s + (3 + 2i)13-s + 14-s + 16-s − 17-s − i·19-s + i·20-s + 22-s − 3·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 0.377i·7-s − 0.353i·8-s + 0.316·10-s − 0.301i·11-s + (0.832 + 0.554i)13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.229i·19-s + 0.223i·20-s + 0.213·22-s − 0.625·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.379306868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379306868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 5 | \( 1 + iT - 5T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 9iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.923151729380885625586443407000, −8.724349818338890100419036863488, −7.56744132373493208535869427739, −7.02866321813685786888235586540, −5.98239029814800595962653552624, −5.41766685014813799512945661417, −4.23352339657263625153491165396, −3.70749095648718184002549111569, −2.06119802347000057699424732207, −0.58334601001825132688012284631,
1.28109676730845854431453883218, 2.48381326839205881934110771994, 3.35105147874108255698222362429, 4.26293531257245778785264389355, 5.33357163356571344567471986890, 6.12366523540809836145610259230, 7.07276906570607267469539849406, 8.071395067409180812612779582792, 8.715142890245018728523541581837, 9.605415753795048473971187581684
