| L(s) = 1 | + (−2.82 − 2.82i)3-s + (17.6 − 17.6i)5-s − 197. i·7-s − 227. i·9-s + (541. − 541. i)11-s + (−657. − 657. i)13-s − 99.9·15-s + 152.·17-s + (428. + 428. i)19-s + (−557. + 557. i)21-s + 2.10e3i·23-s − 625i·25-s + (−1.32e3 + 1.32e3i)27-s + (−1.31e3 − 1.31e3i)29-s + 8.01e3·31-s + ⋯ |
| L(s) = 1 | + (−0.181 − 0.181i)3-s + (0.316 − 0.316i)5-s − 1.52i·7-s − 0.934i·9-s + (1.34 − 1.34i)11-s + (−1.07 − 1.07i)13-s − 0.114·15-s + 0.128·17-s + (0.271 + 0.271i)19-s + (−0.276 + 0.276i)21-s + 0.829i·23-s − 0.200i·25-s + (−0.350 + 0.350i)27-s + (−0.289 − 0.289i)29-s + 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.884585339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884585339\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-17.6 + 17.6i)T \) |
| good | 3 | \( 1 + (2.82 + 2.82i)T + 243iT^{2} \) |
| 7 | \( 1 + 197. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-541. + 541. i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (657. + 657. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 - 152.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-428. - 428. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 - 2.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (1.31e3 + 1.31e3i)T + 2.05e7iT^{2} \) |
| 31 | \( 1 - 8.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (4.17e3 - 4.17e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 6.83e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (367. - 367. i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 - 6.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (8.83e3 - 8.83e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (-2.36e4 + 2.36e4i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (-3.46e4 - 3.46e4i)T + 8.44e8iT^{2} \) |
| 67 | \( 1 + (-1.28e4 - 1.28e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.67e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 4.14e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.06e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-4.91e4 - 4.91e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 6.26e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 2.78e4T + 8.58e9T^{2} \) |
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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
−10.25580704336590270385625350781, −9.603090488243850190299985637994, −8.474105248192037718758576486815, −7.38701086670468640937597325582, −6.49912576734050583251931484064, −5.51708180794387918195809789437, −4.08890134100607489467807524926, −3.23884525681282585480855210347, −1.14478842879130572035292705661, −0.56849097047255577533675387634,
1.84746406005028203506615317322, 2.52805278997219675883513293336, 4.39245629632767019836866387185, 5.17569141893187999581344800056, 6.40890929138591121344855610183, 7.21326809371299529252296715853, 8.583754281858923163389855196094, 9.470653908510707880550594118128, 10.05684017579594550860330005318, 11.46084446859279031589879418445
