L(s) = 1 | + (4.64 + 8.04i)3-s + (−17.1 − 9.92i)5-s + (−18.3 + 2.13i)7-s + (−29.6 + 51.4i)9-s + (−10.2 + 5.91i)11-s + 19.9i·13-s − 184. i·15-s + (1.57 − 0.908i)17-s + (−3.66 + 6.34i)19-s + (−102. − 138. i)21-s + (92.2 + 53.2i)23-s + (134. + 233. i)25-s − 300.·27-s − 191.·29-s + (62.5 + 108. i)31-s + ⋯ |
L(s) = 1 | + (0.894 + 1.54i)3-s + (−1.53 − 0.887i)5-s + (−0.993 + 0.115i)7-s + (−1.09 + 1.90i)9-s + (−0.280 + 0.162i)11-s + 0.426i·13-s − 3.17i·15-s + (0.0224 − 0.0129i)17-s + (−0.0442 + 0.0765i)19-s + (−1.06 − 1.43i)21-s + (0.835 + 0.482i)23-s + (1.07 + 1.86i)25-s − 2.14·27-s − 1.22·29-s + (0.362 + 0.627i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0114i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00417908 + 0.732051i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00417908 + 0.732051i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (18.3 - 2.13i)T \) |
good | 3 | \( 1 + (-4.64 - 8.04i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (17.1 + 9.92i)T + (62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (10.2 - 5.91i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 19.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-1.57 + 0.908i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (3.66 - 6.34i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-92.2 - 53.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-62.5 - 108. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (158. - 274. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 321. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 74.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-77.2 + 133. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-159. - 276. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (30.5 + 52.9i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (267. + 154. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (514. - 297. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 48.6iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (667. - 385. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-831. - 480. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (567. + 327. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 704. iT - 9.12e5T^{2} \) |
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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
−13.75616130175757433348401968034, −12.61229575943822387251150536482, −11.48675762059825336507450311330, −10.32701186032910541041892801667, −9.196329361142904846321781153898, −8.627934330060544663863904877811, −7.40760233601107401263343984553, −5.13206548786430281008932886880, −4.06937633944397133528100345573, −3.21776443116134806282382032703,
0.34627046476442821770211882259, 2.73241138990969254474995474914, 3.61618886731582318603448738050, 6.34990238834157539277149446430, 7.28365641080867673293279573933, 7.85948314764237528927808579933, 9.029894599750074064559706980452, 10.72668909073057083399677689462, 11.86856607455127842771520321045, 12.72355729378533496802702366843