L(s) = 1 | + (0.959 + 0.281i)2-s + (1.01 − 1.16i)3-s + (0.841 + 0.540i)4-s + (0.324 − 2.25i)5-s + (1.30 − 0.835i)6-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (0.0867 + 0.603i)9-s + (0.945 − 2.07i)10-s + (−1.30 + 0.382i)11-s + (1.48 − 0.435i)12-s + (0.0366 − 0.0801i)13-s + (−0.142 − 0.989i)14-s + (−2.30 − 2.66i)15-s + (0.415 + 0.909i)16-s + (1.63 − 1.05i)17-s + ⋯ |
L(s) = 1 | + (0.678 + 0.199i)2-s + (0.584 − 0.674i)3-s + (0.420 + 0.270i)4-s + (0.144 − 1.00i)5-s + (0.531 − 0.341i)6-s + (−0.157 − 0.343i)7-s + (0.231 + 0.267i)8-s + (0.0289 + 0.201i)9-s + (0.299 − 0.654i)10-s + (−0.392 + 0.115i)11-s + (0.428 − 0.125i)12-s + (0.0101 − 0.0222i)13-s + (−0.0380 − 0.264i)14-s + (−0.595 − 0.686i)15-s + (0.103 + 0.227i)16-s + (0.397 − 0.255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12490 - 0.771541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12490 - 0.771541i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (3.16 - 3.60i)T \) |
good | 3 | \( 1 + (-1.01 + 1.16i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.324 + 2.25i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (1.30 - 0.382i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.0366 + 0.0801i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.63 + 1.05i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (1.19 + 0.770i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.33 - 0.856i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.52 - 2.91i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.576 - 4.01i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.345 - 2.40i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.425 + 0.490i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 + (0.587 + 1.28i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.104 - 0.227i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.73 + 6.62i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.348 - 0.102i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (15.6 + 4.58i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.64 - 4.27i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.393 - 0.862i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.01 - 7.07i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-7.03 + 8.12i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.400 - 2.78i)T + (-93.0 - 27.3i)T^{2} \) |
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
−11.89989053179996280555056703711, −10.67913410333978166885319493794, −9.539849556471166644874576510020, −8.393461100039946859977045897650, −7.73405116210861912003917355853, −6.72895406799167805247983118179, −5.42562657557406214866903539971, −4.53674861836790713676030725887, −3.05998613223561165622417772253, −1.59666220529050133095844519986,
2.44583502382680616283163680809, 3.34985778389422224737299907686, 4.37810196037198510988641885324, 5.82365922098551147580605063488, 6.65410303415971771256001048323, 7.924936190455396366760242902817, 9.093039627521807941884489313591, 10.12959191418483112325698141837, 10.62506776558567537776199481124, 11.76155977290122335450387819815