| L(s) = 1 | − 2.63·5-s − 12.5i·7-s + 5.79i·11-s − 8.78·13-s + 30.1·17-s − 17.4i·19-s + 2.48i·23-s − 18.0·25-s − 26.4·29-s − 38.0i·31-s + 33.0i·35-s + 47.7·37-s − 53.3·41-s + 30.7i·43-s + 16.2i·47-s + ⋯ |
| L(s) = 1 | − 0.527·5-s − 1.79i·7-s + 0.527i·11-s − 0.675·13-s + 1.77·17-s − 0.916i·19-s + 0.108i·23-s − 0.722·25-s − 0.910·29-s − 1.22i·31-s + 0.945i·35-s + 1.29·37-s − 1.30·41-s + 0.714i·43-s + 0.345i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5720655079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5720655079\) |
| \(L(2)\) |
|
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 | \( 1 + 2.63T + 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 - 5.79iT - 121T^{2} \) |
| 13 | \( 1 + 8.78T + 169T^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 + 17.4iT - 361T^{2} \) |
| 23 | \( 1 - 2.48iT - 529T^{2} \) |
| 29 | \( 1 + 26.4T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 - 47.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 62.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
| 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
−9.546375397745465787829559503913, −7.983012467655977881738217986937, −7.57079546318208884868896232899, −7.01073800816261838656152017492, −5.79255128492677430564714211140, −4.61295734718219722490922789393, −4.01325823272254655734233135246, −2.99954269148978050167109955778, −1.36065485529664368040363534941, −0.17647321890392458159850254574,
1.67740910216012659589016563922, 2.89187202681990831981416516946, 3.69313545539381905100733838704, 5.19750433985557393778573056114, 5.60289090388782454453954633981, 6.58156767651392955463999990182, 7.935117434994991404175968815206, 8.137688045246647092976128981946, 9.278336216705135077788574854987, 9.799923820812078952211905942789
