L(s) = 1 | + 3.32·5-s − 4.21·7-s + 16.0·11-s + 17.6i·13-s + 4.44i·17-s + 26.5i·19-s − 5.59i·23-s − 13.9·25-s − 50.0·29-s + 11.4·31-s − 14.0·35-s − 24.5i·37-s − 18.5i·41-s − 37.2i·43-s + 73.5i·47-s + ⋯ |
L(s) = 1 | + 0.664·5-s − 0.602·7-s + 1.46·11-s + 1.35i·13-s + 0.261i·17-s + 1.39i·19-s − 0.243i·23-s − 0.558·25-s − 1.72·29-s + 0.370·31-s − 0.400·35-s − 0.664i·37-s − 0.453i·41-s − 0.865i·43-s + 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.659222745\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.659222745\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.32T + 25T^{2} \) |
| 7 | \( 1 + 4.21T + 49T^{2} \) |
| 11 | \( 1 - 16.0T + 121T^{2} \) |
| 13 | \( 1 - 17.6iT - 169T^{2} \) |
| 17 | \( 1 - 4.44iT - 289T^{2} \) |
| 19 | \( 1 - 26.5iT - 361T^{2} \) |
| 23 | \( 1 + 5.59iT - 529T^{2} \) |
| 29 | \( 1 + 50.0T + 841T^{2} \) |
| 31 | \( 1 - 11.4T + 961T^{2} \) |
| 37 | \( 1 + 24.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 37.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.32T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 12.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 45.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 3.04T + 5.32e3T^{2} \) |
| 79 | \( 1 + 17.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 92.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 122.T + 9.40e3T^{2} \) |
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
−9.419950038908186636788623232188, −8.845604998678058113859326430136, −7.73597036667607599999893895287, −6.76636073625446434704265703155, −6.23674779895710212947926518358, −5.51744328491972320720920357788, −4.10057481914868612118011355048, −3.71358539299768460748134701037, −2.16802037302832292042138053484, −1.40224154079959418111567882657,
0.42895227081085034884019817234, 1.69094138635176081018447224657, 2.91386214610721253116207627955, 3.71129700572425884371143260742, 4.87279221679391265151512347103, 5.76717163408031627635984691619, 6.45174805209656092565846928288, 7.19438422447604063297894115414, 8.157787079331702142902425143726, 9.180766409165111160701731252169