L(s) = 1 | + (17.2 + 17.2i)3-s + (−154. + 154. i)7-s + 355. i·9-s + 127. i·11-s + (−335. + 335. i)13-s + (1.15e3 + 1.15e3i)17-s + 28.2·19-s − 5.32e3·21-s + (2.78e3 + 2.78e3i)23-s + (−1.93e3 + 1.93e3i)27-s − 3.38e3i·29-s − 5.38e3i·31-s + (−2.20e3 + 2.20e3i)33-s + (−1.15e4 − 1.15e4i)37-s − 1.15e4·39-s + ⋯ |
L(s) = 1 | + (1.10 + 1.10i)3-s + (−1.18 + 1.18i)7-s + 1.46i·9-s + 0.317i·11-s + (−0.549 + 0.549i)13-s + (0.969 + 0.969i)17-s + 0.0179·19-s − 2.63·21-s + (1.09 + 1.09i)23-s + (−0.511 + 0.511i)27-s − 0.748i·29-s − 1.00i·31-s + (−0.352 + 0.352i)33-s + (−1.38 − 1.38i)37-s − 1.21·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.691725743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.691725743\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + (-17.2 - 17.2i)T + 243iT^{2} \) |
| 7 | \( 1 + (154. - 154. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 127. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (335. - 335. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.15e3 - 1.15e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 28.2T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.78e3 - 2.78e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 3.38e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.38e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.15e4 + 1.15e4i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.43e3 - 1.43e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-219. + 219. i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.27e4 - 2.27e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 2.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.43e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.89e4 + 2.89e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 2.41e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.85e4 - 2.85e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 2.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.89e4 - 1.89e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 8.17e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-7.67e4 - 7.67e4i)T + 8.58e9iT^{2} \) |
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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
−10.67362642392215354382186258509, −9.602460053389161732613628953522, −9.451036629455313888193992138312, −8.547630599330800105271304470654, −7.46240824154378365471261852167, −6.12271357407724059384550664955, −5.09887808363962835988972637390, −3.80633855567751915823463988239, −3.08571137322955083124840162511, −2.05004837482403233266245740876,
0.34848590167076951132319722845, 1.31604730590091623918576145527, 3.00224284603275603648535001666, 3.28817686927100302989038252398, 5.02699056791739881001917887584, 6.72783681320703935474451244564, 7.00522237241801538644267241086, 7.974225076720282224332775003354, 8.879706881738638102618324793523, 9.874356634069636523414323114768