L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (−0.866 − 0.499i)25-s + (0.707 + 1.22i)26-s + (−0.866 − 0.499i)32-s − 1.41·34-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (−0.866 − 0.499i)25-s + (0.707 + 1.22i)26-s + (−0.866 − 0.499i)32-s − 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.698306124\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.698306124\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−10.09668256403529590553036571258, −9.339961940217215710522070140503, −8.590576557424563564510420489027, −7.26936529084338686246055238888, −6.53997326886289478204797337404, −5.39359275853674283116118357473, −4.65137667267739863832428856911, −4.08533714243369585529571767341, −2.43702674696240688646207824865, −1.56997585376123356412284055248,
2.20521203852142335097996614980, 3.31666910974646657766852397326, 4.03137258369416807580917582221, 5.31501265359363504375946607783, 6.16827925347434225808974143010, 6.80238051486425567189278622551, 7.57719904095778612843745925885, 8.512280805834303309378364583846, 9.557920937791690140886301999819, 10.65429995339887270788488095082