L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)11-s + (−0.5 − 0.866i)13-s − i·17-s + 1.41·19-s + (0.965 + 0.258i)23-s + (−0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (−0.366 − 1.36i)29-s + (0.258 − 0.965i)31-s + 33-s + (−0.965 + 0.258i)39-s + (0.707 − 1.22i)43-s + (−1.22 − 0.707i)47-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)11-s + (−0.5 − 0.866i)13-s − i·17-s + 1.41·19-s + (0.965 + 0.258i)23-s + (−0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (−0.366 − 1.36i)29-s + (0.258 − 0.965i)31-s + 33-s + (−0.965 + 0.258i)39-s + (0.707 − 1.22i)43-s + (−1.22 − 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.228642559\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.228642559\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−8.976072628031187888966009077063, −7.81231857828688418956514911329, −7.52502902311154093311047792475, −6.85960926512929577887386799775, −5.80370642049038728211485783910, −5.18395458626706858645274693284, −4.02320767574803571064703607440, −2.89853803815336266780233576010, −2.19329946921728151922679222435, −0.839303937749535795864869331858,
1.56667907659407363478024157109, 3.02976255108248284536375492040, 3.53318308385202455099714143663, 4.57466085606981794688632288684, 5.28574313400517720094909338613, 6.09589999543846328343825614201, 7.04614168232612921646207581570, 7.973564594543358099751414722016, 8.707570970834722437176709173742, 9.361365375507704592507310814361