L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + 1.93i·26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + 1.93i·26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8355299361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8355299361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 41 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.93T + T^{2} \) |
| 97 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.304740275664754218788024836435, −8.887187359021881249393629806634, −7.996327798648295772361188084775, −7.20528156483661560622247204300, −6.29833154706224157698784039949, −5.61771033439227744694634092171, −4.85124731266800994109331965306, −3.11437584173295641246290113908, −2.26238296096108994693738257683, −0.918785315892617106326793551235,
1.67719598762745788912517095488, 2.20683038634058513972699372644, 3.57106730456090437439555961658, 4.58268216500588272182625492197, 6.17953680907566618258237410787, 6.37354753162550699366786992545, 7.28147449679951386022492170156, 8.357403317411204033660568167858, 9.080676676698046781978875771712, 9.487068460630137538867667127125