| L(s) = 1 | + (2.28 − 1.66i)2-s + (−1.11 + 1.53i)3-s + (2.47 − 7.60i)4-s − 47.5·5-s + 5.35i·6-s + (−18.0 + 55.6i)7-s + (−6.99 − 21.5i)8-s + (23.9 + 73.6i)9-s + (−108. + 79.0i)10-s + (−124. − 40.3i)11-s + (8.90 + 12.2i)12-s + (136. − 187. i)13-s + (51.1 + 157. i)14-s + (52.9 − 72.8i)15-s + (−51.7 − 37.6i)16-s + (−370. + 120. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.123 + 0.170i)3-s + (0.154 − 0.475i)4-s − 1.90·5-s + 0.148i·6-s + (−0.369 + 1.13i)7-s + (−0.109 − 0.336i)8-s + (0.295 + 0.908i)9-s + (−1.08 + 0.790i)10-s + (−1.02 − 0.333i)11-s + (0.0618 + 0.0850i)12-s + (0.805 − 1.10i)13-s + (0.261 + 0.803i)14-s + (0.235 − 0.323i)15-s + (−0.202 − 0.146i)16-s + (−1.28 + 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.111121 + 0.310374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.111121 + 0.310374i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.28 + 1.66i)T \) |
| 31 | \( 1 + (886. + 371. i)T \) |
| good | 3 | \( 1 + (1.11 - 1.53i)T + (-25.0 - 77.0i)T^{2} \) |
| 5 | \( 1 + 47.5T + 625T^{2} \) |
| 7 | \( 1 + (18.0 - 55.6i)T + (-1.94e3 - 1.41e3i)T^{2} \) |
| 11 | \( 1 + (124. + 40.3i)T + (1.18e4 + 8.60e3i)T^{2} \) |
| 13 | \( 1 + (-136. + 187. i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (370. - 120. i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (22.9 - 16.6i)T + (4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 + (160. - 52.2i)T + (2.26e5 - 1.64e5i)T^{2} \) |
| 29 | \( 1 + (-32.6 - 44.9i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 37 | \( 1 + 844. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (807. - 586. i)T + (8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + (-1.32e3 - 1.82e3i)T + (-1.05e6 + 3.25e6i)T^{2} \) |
| 47 | \( 1 + (-2.33e3 - 1.69e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (2.48e3 - 807. i)T + (6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-1.95e3 - 1.41e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 - 303. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 6.98e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-2.30e3 - 7.09e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (8.51e3 + 2.76e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (-4.26e3 + 1.38e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (199. + 273. i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 + (1.13e4 + 3.68e3i)T + (5.07e7 + 3.68e7i)T^{2} \) |
| 97 | \( 1 + (1.63e3 - 5.04e3i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−15.02061954405232626541582376096, −13.17644120946957167735935601611, −12.52400610099075454346370186391, −11.24820160334993511480855725856, −10.69118488919654202743531378008, −8.642522481559145733335951755435, −7.64913922341593227005340866424, −5.70570649681859430247734264283, −4.30994179653508197567282794962, −2.87110211274764393123264752879,
0.15013458114243050187257784638, 3.65499650640032219718377348415, 4.43377200765891005767674653927, 6.78176130921124573509168348389, 7.37692399917943798639976295305, 8.761644012742647077943618913573, 10.75517410187956429664625569901, 11.71779787199509106204893710334, 12.71985661719264339445212636681, 13.77187196991419615339237072817
