| L(s) = 1 | + 0.890i·5-s + 1.60i·7-s + 2.49i·11-s + (0.109 + 3.60i)13-s − 2·17-s − 1.60i·19-s − 4.98·23-s + 4.20·25-s − 6.98·29-s + 3.38i·31-s − 1.42·35-s − 0.987i·37-s + 2.67i·41-s − 8.19·43-s − 1.50i·47-s + ⋯ |
| L(s) = 1 | + 0.398i·5-s + 0.606i·7-s + 0.751i·11-s + (0.0304 + 0.999i)13-s − 0.485·17-s − 0.367i·19-s − 1.04·23-s + 0.841·25-s − 1.29·29-s + 0.607i·31-s − 0.241·35-s − 0.162i·37-s + 0.417i·41-s − 1.24·43-s − 0.219i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6874010659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6874010659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.109 - 3.60i)T \) |
| good | 5 | \( 1 - 0.890iT - 5T^{2} \) |
| 7 | \( 1 - 1.60iT - 7T^{2} \) |
| 11 | \( 1 - 2.49iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.60iT - 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 3.38iT - 31T^{2} \) |
| 37 | \( 1 + 0.987iT - 37T^{2} \) |
| 41 | \( 1 - 2.67iT - 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 + 1.50iT - 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 0.713iT - 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 7.82iT - 67T^{2} \) |
| 71 | \( 1 + 9.70iT - 71T^{2} \) |
| 73 | \( 1 + 1.78iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 + 9.78iT - 97T^{2} \) |
| 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
−8.936977372079528676037085916137, −8.214158203229941196706351128185, −7.24427306280046830449652836794, −6.76077468434718419004672436213, −5.99983171916765658040930052490, −5.07038931446097852641260832634, −4.35762389528447426470863889855, −3.43935713198977570464192214744, −2.37670693521659541626685279170, −1.70012650693286269923739745557,
0.19808789589761748558332515654, 1.29639447347440131792161893753, 2.52771134159260315480789015374, 3.56941144950199183545293975992, 4.17766729311554390737070306915, 5.24844235909539550320321102064, 5.79182829025462380512309310047, 6.67643659909258239705119445257, 7.49754607494805207095804884938, 8.199738096404279932928476522606
