| L(s) = 1 | + (−2.35 − 3.23i)2-s + (−11.7 − 3.82i)3-s + (−4.94 + 15.2i)4-s + (55.5 + 6.43i)5-s + (15.3 + 47.1i)6-s + 133. i·7-s + (60.8 − 19.7i)8-s + (−72.5 − 52.7i)9-s + (−109. − 194. i)10-s + (434. − 315. i)11-s + (116. − 160. i)12-s + (322. − 443. i)13-s + (433. − 314. i)14-s + (−629. − 288. i)15-s + (−207. − 150. i)16-s + (741. − 240. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.755 − 0.245i)3-s + (−0.154 + 0.475i)4-s + (0.993 + 0.115i)5-s + (0.173 + 0.534i)6-s + 1.03i·7-s + (0.336 − 0.109i)8-s + (−0.298 − 0.216i)9-s + (−0.346 − 0.616i)10-s + (1.08 − 0.786i)11-s + (0.233 − 0.321i)12-s + (0.528 − 0.727i)13-s + (0.590 − 0.429i)14-s + (−0.722 − 0.330i)15-s + (−0.202 − 0.146i)16-s + (0.621 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.06827 - 0.617088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06827 - 0.617088i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-55.5 - 6.43i)T \) |
| good | 3 | \( 1 + (11.7 + 3.82i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 133. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-434. + 315. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-322. + 443. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-741. + 240. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (582. + 1.79e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.82e3 - 2.51e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.64e3 + 8.12e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.29e3 - 3.99e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (724. - 996. i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-3.30e3 - 2.39e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 46.6iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.21e4 - 3.93e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.40e4 + 7.80e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.11e4 + 1.53e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.25e4 + 2.36e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-9.11e3 + 2.96e3i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.33e4 - 7.19e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.31e4 + 1.81e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-7.02e3 + 2.16e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (7.42e4 - 2.41e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (9.85e4 - 7.16e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.09e5 + 3.55e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.14910709203726227385073858701, −12.95947191746618294212798200475, −11.80688435178350572829435901078, −11.03562462020973562207032268752, −9.530791940146120626275652045011, −8.621786647141550197486916639266, −6.45859310108662832970924094701, −5.49774032821137755560740443920, −2.92250236247331677077253382320, −0.996799790475568867365821450236,
1.31274866094372593690910363901, 4.46483600688137384510719428658, 5.95808787965607361646801747233, 6.96504734884267833283435304644, 8.765416031152821427024647118980, 10.04630534114158407373491590329, 10.84921878337122438178640213132, 12.41976151819295073952553849789, 13.96758861823067593296081731557, 14.55384494468960519261603325096
