L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.366 − 1.36i)11-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (1 + 0.999i)22-s + (0.866 + 0.5i)25-s + (−1 − i)29-s + (0.866 + 0.499i)32-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (1 − i)43-s + (−1.36 − 0.366i)44-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.366 − 1.36i)11-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (1 + 0.999i)22-s + (0.866 + 0.5i)25-s + (−1 − i)29-s + (0.866 + 0.499i)32-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (1 − i)43-s + (−1.36 − 0.366i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6811935923\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6811935923\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.38800795207680026420722055282, −9.465348940767392261054246308764, −8.836552815379729109883448809784, −7.898720415132344780431241107515, −7.16637269554654030052435671516, −6.18602387845473621125474620743, −5.48043819814346753942083354044, −4.11927380086390421294921127168, −2.70685575767388143869607669301, −1.07685551885953780678599097414,
1.59722733950706299319784831091, 2.63733231733156691773281628758, 4.05577511815296324679240997936, 4.94768704561121249012856496663, 6.50533968112691678353395698591, 7.38697182393370878027485882134, 7.85822884677024023270802568100, 9.062138602893218024396096426673, 9.715605008923929359444889611746, 10.46576761845601893373361780041