L(s) = 1 | + (3.78 − 2.18i)2-s + (3.78 − 3.55i)3-s + (5.55 − 9.62i)4-s + (6.55 − 21.7i)6-s + (10.4 − 6.05i)7-s − 13.6i·8-s + (1.67 − 26.9i)9-s + (−5.01 − 8.67i)11-s + (−13.2 − 56.2i)12-s + (−42.0 − 24.2i)13-s + (26.4 − 45.8i)14-s + (14.6 + 25.4i)16-s + 75.3i·17-s + (−52.5 − 105. i)18-s + 116.·19-s + ⋯ |
L(s) = 1 | + (1.33 − 0.772i)2-s + (0.728 − 0.684i)3-s + (0.694 − 1.20i)4-s + (0.446 − 1.48i)6-s + (0.566 − 0.327i)7-s − 0.602i·8-s + (0.0620 − 0.998i)9-s + (−0.137 − 0.237i)11-s + (−0.317 − 1.35i)12-s + (−0.897 − 0.518i)13-s + (0.505 − 0.875i)14-s + (0.229 + 0.397i)16-s + 1.07i·17-s + (−0.688 − 1.38i)18-s + 1.40·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.66166 - 3.80990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.66166 - 3.80990i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.78 + 3.55i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-3.78 + 2.18i)T + (4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-10.4 + 6.05i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (5.01 + 8.67i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (42.0 + 24.2i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 75.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (32.9 + 19.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (11.3 + 19.5i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (15.0 - 26.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 130. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (173. - 300. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (23.1 - 13.3i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-399. + 230. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 438. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-4.18 + 7.24i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-41.0 - 71.0i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (591. + 341. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 470. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-243. - 420. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (85.8 - 49.5i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 8.80T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-572. + 330. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.92761421759883169616243394622, −10.85164440285160373177544639366, −9.799345613772331272728321076995, −8.334369620119237728205321916987, −7.48632247731676724074519418118, −6.08840387784653910378577041042, −4.92477112665501052862373087042, −3.67681689799754398261692640164, −2.65686890630648103745891660536, −1.36811465202354043543255087298,
2.45551850029203095726292709063, 3.71041751318991279438806323193, 4.87706792021699482627991481309, 5.39499840253317081505826602935, 7.09085795166116304952498118079, 7.75948179726150177653138145147, 9.142413087718551160207709769630, 9.985094921689026075966694213568, 11.47242401899242172988520798092, 12.25524293558349176151861087560