| L(s) = 1 | − 1.03i·5-s + i·7-s + 0.378·11-s − 2·13-s + 3.86i·17-s − 1.46i·19-s − 5.27·23-s + 3.92·25-s − 3.48i·29-s + 2.53i·31-s + 1.03·35-s − 2.92·37-s + 8.76i·41-s − 4i·43-s − 8.48·47-s + ⋯ |
| L(s) = 1 | − 0.462i·5-s + 0.377i·7-s + 0.114·11-s − 0.554·13-s + 0.937i·17-s − 0.335i·19-s − 1.10·23-s + 0.785·25-s − 0.647i·29-s + 0.455i·31-s + 0.174·35-s − 0.481·37-s + 1.36i·41-s − 0.609i·43-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4733625824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4733625824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.03iT - 5T^{2} \) |
| 11 | \( 1 - 0.378T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 1.46iT - 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 + 3.48iT - 29T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 8.76iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 8.53iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 1.03iT - 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.494650518432491597501263637159, −8.283015970709423336394244945838, −7.27026937983668489406595188281, −6.49084264969006502600496686161, −5.78085661923982013432472799746, −4.96269211652569372922987070681, −4.28726185801003041537927826548, −3.31947394032537605375467319529, −2.32011793205413141339681012209, −1.35873983513891502876849621757,
0.13198158464834490890815825075, 1.57018560794972513846132592052, 2.64979457723044386617528102893, 3.42568263243295025944483309618, 4.36528220404601243380546598004, 5.10499113389231277479186172093, 5.99845323791450624806251889889, 6.78326140310516385154817518979, 7.38429334939684247835328427124, 8.006658475750344919869069534062
