L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.258 + 0.965i)6-s + (1.36 − 0.366i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.965 − 0.258i)11-s + 12-s − 13-s − 1.41i·14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.258 + 0.965i)6-s + (1.36 − 0.366i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.965 − 0.258i)11-s + 12-s − 13-s − 1.41i·14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4707429310\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4707429310\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.606404325328078076280078527699, −7.74253910493803447978632916553, −7.23477823995353846804538745016, −5.88506390708495542523775129021, −5.30681411391830999765553308771, −4.48389256967469654764079772769, −4.01235593345948518495435864232, −2.93678212993564131154870241359, −1.88909766697569953836686135801, −0.31420132406753322645061261569,
1.33918142391637270162777268172, 2.74171161085315192774049187138, 4.33188117138754069248243289630, 4.95021166415069182723565140005, 5.11392878392539261780115502184, 6.08956284238718456969046338492, 7.22316979628490631794956714082, 7.47783673526663050116984820286, 8.376118489067497418891577581331, 8.606208960069043888123196510592