| L(s) = 1 | − 2·3-s + (−3 + 3i)7-s + 9-s + (1 + i)11-s + 2i·13-s + (−1 + i)17-s + (3 + 3i)19-s + (6 − 6i)21-s + (−1 − i)23-s + 4·27-s + (−7 + 7i)29-s − 2i·31-s + (−2 − 2i)33-s − 6i·37-s − 4i·39-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + (−1.13 + 1.13i)7-s + 0.333·9-s + (0.301 + 0.301i)11-s + 0.554i·13-s + (−0.242 + 0.242i)17-s + (0.688 + 0.688i)19-s + (1.30 − 1.30i)21-s + (−0.208 − 0.208i)23-s + 0.769·27-s + (−1.29 + 1.29i)29-s − 0.359i·31-s + (−0.348 − 0.348i)33-s − 0.986i·37-s − 0.640i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 + (7 - 7i)T - 29iT^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + (3 - 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-11 + 11i)T - 97iT^{2} \) |
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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
−9.238942270210354711088330763436, −8.532386326098033075536076699371, −7.22255523085825466519027665419, −6.56719111629516412301280577319, −5.75715420349267141210556490114, −5.37856479852763345216332433780, −4.11119843435503972832553631872, −3.07283504649750636870310439265, −1.80318515915811479115274602692, 0,
0.992545958293620806338961011892, 2.90384284904392945037127602770, 3.81070009470153480989315449889, 4.80692827445366092158236224068, 5.74396736749512175495356481205, 6.41390294988502804524204609402, 7.07867655731667953341167169694, 7.896187354212271565040723769402, 9.101139535687888154757343634177
