L(s) = 1 | + (−1 − i)3-s + (−2 + 2i)5-s + 4i·7-s − i·9-s + (−1 + i)11-s + (2 + 2i)13-s + 4·15-s + 4·17-s + (−5 − 5i)19-s + (4 − 4i)21-s − 4i·23-s − 3i·25-s + (−4 + 4i)27-s + (−6 − 6i)29-s − 8·31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.577i)3-s + (−0.894 + 0.894i)5-s + 1.51i·7-s − 0.333i·9-s + (−0.301 + 0.301i)11-s + (0.554 + 0.554i)13-s + 1.03·15-s + 0.970·17-s + (−1.14 − 1.14i)19-s + (0.872 − 0.872i)21-s − 0.834i·23-s − 0.600i·25-s + (−0.769 + 0.769i)27-s + (−1.11 − 1.11i)29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (5 + 5i)T + 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (6 + 6i)T + 29iT^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 2i)T - 37iT^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (2 - 2i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3 + 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6 - 6i)T + 61iT^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (7 + 7i)T + 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.447651134227729078341170471500, −8.755096790324712371861479237440, −7.77891682488063234786745697551, −6.96779062635909559457398390557, −6.23097237621964385416743819899, −5.52106267555180265475607362931, −4.19942307312353317363870580401, −3.09047139588442463285711751689, −2.02325812911198847053479850855, 0,
1.37179631026388951266522490264, 3.70290815934971434515936580770, 3.94921456251181557653383699816, 5.11623368886745793290367160927, 5.72812836812556639272719159386, 7.16217914091863556617504823248, 7.88737052994389186279896323492, 8.435780683320674384609438935994, 9.689360619551227449509741235991