L(s) = 1 | + (2.80 − 0.325i)2-s + (7.78 − 1.83i)4-s + 18.5i·5-s + 9.32·7-s + (21.2 − 7.68i)8-s + (6.04 + 52.0i)10-s − 39.7i·11-s − 32.9i·13-s + (26.2 − 3.04i)14-s + (57.2 − 28.5i)16-s − 90.5·17-s + 72.5i·19-s + (33.9 + 144. i)20-s + (−12.9 − 111. i)22-s − 45.3·23-s + ⋯ |
L(s) = 1 | + (0.993 − 0.115i)2-s + (0.973 − 0.228i)4-s + 1.65i·5-s + 0.503·7-s + (0.940 − 0.339i)8-s + (0.191 + 1.64i)10-s − 1.08i·11-s − 0.703i·13-s + (0.500 − 0.0580i)14-s + (0.895 − 0.445i)16-s − 1.29·17-s + 0.876i·19-s + (0.379 + 1.61i)20-s + (−0.125 − 1.08i)22-s − 0.411·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.60417 + 0.455759i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60417 + 0.455759i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.80 + 0.325i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 18.5iT - 125T^{2} \) |
| 7 | \( 1 - 9.32T + 343T^{2} \) |
| 11 | \( 1 + 39.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 143. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.77iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 241. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 149. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 508. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 950. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 175. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 158.T + 9.12e5T^{2} \) |
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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
−14.13520928965667039850235907233, −13.45534945045106070492612877621, −11.85487696013106346625473796043, −10.98830124648593787073928448217, −10.29202774178244885596600016187, −8.045645377179463290449833295468, −6.75563828218632858940465637041, −5.74378298449712357146918945872, −3.81609962226723954690759502243, −2.50770158075500088906091389333,
1.83479725650786765735280718315, 4.42364770896817431671164723413, 4.98395312422848656000627081565, 6.73041038624901430387064650907, 8.190377517463941256120547409983, 9.399296279873317936535561392385, 11.16870265013637008373210004521, 12.19756373989451988137867418350, 12.95864787075364727770179031832, 13.88576627261405933574356304903