| L(s) = 1 | + 3-s + 2i·5-s − 2i·7-s + 9-s + (3 − 2i)13-s + 2i·15-s − 2·17-s + 6i·19-s − 2i·21-s − 4·23-s + 25-s + 27-s + 10·29-s + 10i·31-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894i·5-s − 0.755i·7-s + 0.333·9-s + (0.832 − 0.554i)13-s + 0.516i·15-s − 0.485·17-s + 1.37i·19-s − 0.436i·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s + 1.85·29-s + 1.79i·31-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328844774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328844774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.669029149408725192209302403534, −8.386791930404909916333833681284, −7.41278339185121496125920940011, −6.77894512551745001712123132310, −6.10915943981464401815743566984, −4.98453424994690680497479421768, −3.86112763593529865221040581868, −3.39332894661611031744321486052, −2.36411447835888112330154150030, −1.12805404260112906456073730735,
0.865190569043465432791976012180, 2.13213419224769988985383921591, 2.91639152398017264802031822777, 4.27421890699668819415846303902, 4.62503532348734583399201605139, 5.82750814960662168044895062646, 6.42263811701478597611579039304, 7.49330045732221864983296555148, 8.291205513783373362182260442060, 8.979849600839960952138747224672
