L(s) = 1 | + (1 − i)3-s + (−2 − i)5-s + (1 − i)7-s + i·9-s − 6·11-s + (−1 − i)13-s + (−3 + i)15-s + (1 + i)17-s + 4i·19-s − 2i·21-s + (−5 − 5i)23-s + (3 + 4i)25-s + (4 + 4i)27-s − 8·29-s + 2i·31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.894 − 0.447i)5-s + (0.377 − 0.377i)7-s + 0.333i·9-s − 1.80·11-s + (−0.277 − 0.277i)13-s + (−0.774 + 0.258i)15-s + (0.242 + 0.242i)17-s + 0.917i·19-s − 0.436i·21-s + (−1.04 − 1.04i)23-s + (0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s − 1.48·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (5 + 5i)T + 23iT^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−8.794188448305710897333604456014, −8.133273781263666787893429570413, −7.74666858282848870183266240571, −7.13603628165564361993561849271, −5.63598616533701397511612489279, −4.92356301998507893371402174266, −3.88913050199082126997352725817, −2.79084740052469048219382801277, −1.70840518454198215734051663630, 0,
2.29841520090826336267948251892, 3.15407448129776687345863716947, 4.03514262245889950531270050276, 4.98954626051246393693336769864, 5.87003305866710610881116133294, 7.26770062584142457378537142427, 7.68733416719636164762099603564, 8.552562419724981086689080107769, 9.313708506293157059601807461938