| L(s) = 1 | + (2.35 + 3.23i)2-s + (22.5 + 7.32i)3-s + (−4.94 + 15.2i)4-s + (−54.0 − 14.0i)5-s + (29.3 + 90.2i)6-s + 229. i·7-s + (−60.8 + 19.7i)8-s + (258. + 187. i)9-s + (−81.5 − 208. i)10-s + (222. − 161. i)11-s + (−223. + 307. i)12-s + (121. − 166. i)13-s + (−742. + 539. i)14-s + (−1.11e3 − 714. i)15-s + (−207. − 150. i)16-s + (1.46e3 − 474. i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (1.44 + 0.470i)3-s + (−0.154 + 0.475i)4-s + (−0.967 − 0.252i)5-s + (0.332 + 1.02i)6-s + 1.77i·7-s + (−0.336 + 0.109i)8-s + (1.06 + 0.773i)9-s + (−0.257 − 0.658i)10-s + (0.554 − 0.402i)11-s + (−0.447 + 0.615i)12-s + (0.198 − 0.273i)13-s + (−1.01 + 0.735i)14-s + (−1.28 − 0.819i)15-s + (−0.202 − 0.146i)16-s + (1.22 − 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.67921 + 2.14881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67921 + 2.14881i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.35 - 3.23i)T \) |
| 5 | \( 1 + (54.0 + 14.0i)T \) |
| good | 3 | \( 1 + (-22.5 - 7.32i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 229. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-222. + 161. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-121. + 166. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.46e3 + 474. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-557. - 1.71e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.67e3 + 3.68e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-708. + 2.18e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (973. + 2.99e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.12e3 + 2.91e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.56e4 - 1.13e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 7.88e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.38e3 - 449. i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-4.37e3 - 1.42e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.23e4 + 8.95e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.74e4 + 1.99e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (3.11e4 - 1.01e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.20e4 - 6.80e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.56e4 - 2.14e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.82e3 - 8.69e3i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (4.70e4 - 1.53e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-6.02e4 + 4.38e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (6.47e4 + 2.10e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.75300507888491787617489076444, −14.31902223558419462176574861403, −12.62511550064324065084448433432, −11.80331937149252523493678569241, −9.587614426098925755743144595832, −8.497612633218450435850256282822, −7.948168226559586733021850623824, −5.78699229013353164281670094986, −4.02502038390932677579142478475, −2.79596267331084551503825218576,
1.23749588659124743028264334606, 3.32050060209214033353914396714, 4.12703508766755777244092431773, 7.10259312690547232227751353108, 7.83741168602119240274898820260, 9.423254565117990998493530641163, 10.71848421696746276759811854542, 12.04514949618465174758774248416, 13.35122696424336269386849178926, 14.08832938970074451060282456888
