| L(s) = 1 | − 6.63i·2-s − 12·4-s + (45 − 33.1i)5-s − 59.6i·7-s − 132. i·8-s + (−220. − 298. i)10-s − 252·11-s − 119. i·13-s − 396·14-s − 1.26e3·16-s + 689. i·17-s − 220·19-s + (−540 + 397. i)20-s + 1.67e3i·22-s − 2.43e3i·23-s + ⋯ |
| L(s) = 1 | − 1.17i·2-s − 0.375·4-s + (0.804 − 0.593i)5-s − 0.460i·7-s − 0.732i·8-s + (−0.695 − 0.943i)10-s − 0.627·11-s − 0.195i·13-s − 0.539·14-s − 1.23·16-s + 0.578i·17-s − 0.139·19-s + (−0.301 + 0.222i)20-s + 0.736i·22-s − 0.959i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.564126 - 1.71624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.564126 - 1.71624i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-45 + 33.1i)T \) |
| good | 2 | \( 1 + 6.63iT - 32T^{2} \) |
| 7 | \( 1 + 59.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 + 119. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 689. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 220T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 198T + 1.15e8T^{2} \) |
| 43 | \( 1 + 417. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.82e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5iT - 8.58e9T^{2} \) |
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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
−13.88533960379769354094494703085, −12.98414098550329076800317843724, −12.07099147048482634046765730285, −10.57114513803079156106554651776, −9.964282482124206858491416597958, −8.418048433651762159224634439680, −6.44244651590754007683773395272, −4.57444132457676853726832937226, −2.64897320938473199539532211455, −1.02087537862008371301362993346,
2.49090330731543460257831706405, 5.22676072222744257576886987393, 6.31654591453785310420190729381, 7.48808201075511211134016280125, 8.911885214124723887739297804694, 10.34247707015070465266509454029, 11.73104783609597120170388968097, 13.44870478713871244123565871646, 14.32495061373711937131605361349, 15.37328280401321787132936946656
