L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.965 − 0.258i)11-s + (−1 + i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (0.366 + 1.36i)19-s + i·22-s + (0.707 − 1.22i)23-s + (0.866 − 0.5i)25-s + (1.22 + 0.707i)26-s + 28-s + (0.707 + 0.707i)29-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.965 − 0.258i)11-s + (−1 + i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (0.366 + 1.36i)19-s + i·22-s + (0.707 − 1.22i)23-s + (0.866 − 0.5i)25-s + (1.22 + 0.707i)26-s + 28-s + (0.707 + 0.707i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6801244366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6801244366\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.840898381290618932354167475371, −8.516219990782999831164096846963, −7.34465734650337158928334905203, −6.91358090856614514392169081842, −5.70567610759877775067921719414, −4.77947021721662683205667526932, −4.10663843398362219190898996644, −3.03374515187814304477362518540, −2.49915382717900019090334109769, −1.07117263923990699094627571239,
0.55083658568798211594149295978, 2.47445747780438679738610040398, 3.24879686095199230837471223509, 4.56968582079557652148540632577, 5.34500756562421050614557302917, 5.72092598797106384930399046247, 6.94284812565851042349259296492, 7.24436126070225944008711531183, 8.069482908109743837054918547694, 8.897947668880220211900122366076