| L(s) = 1 | + (0.800 − 1.16i)2-s + (−0.881 + 0.471i)3-s + (−0.718 − 1.86i)4-s + (−1.52 − 1.86i)5-s + (−0.156 + 1.40i)6-s + (−2.50 + 1.67i)7-s + (−2.75 − 0.655i)8-s + (0.555 − 0.831i)9-s + (−3.39 + 0.291i)10-s + (−3.24 + 0.983i)11-s + (1.51 + 1.30i)12-s + (4.36 + 3.58i)13-s + (−0.0533 + 4.26i)14-s + (2.22 + 0.921i)15-s + (−2.96 + 2.68i)16-s + (−3.77 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (0.565 − 0.824i)2-s + (−0.509 + 0.272i)3-s + (−0.359 − 0.933i)4-s + (−0.682 − 0.832i)5-s + (−0.0637 + 0.573i)6-s + (−0.947 + 0.633i)7-s + (−0.972 − 0.231i)8-s + (0.185 − 0.277i)9-s + (−1.07 + 0.0921i)10-s + (−0.977 + 0.296i)11-s + (0.436 + 0.377i)12-s + (1.21 + 0.993i)13-s + (−0.0142 + 1.13i)14-s + (0.574 + 0.237i)15-s + (−0.741 + 0.670i)16-s + (−0.914 + 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0980743 + 0.258321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0980743 + 0.258321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.800 + 1.16i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (1.52 + 1.86i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (2.50 - 1.67i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (3.24 - 0.983i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-4.36 - 3.58i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (3.77 - 1.56i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.446 + 4.53i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (6.02 - 1.19i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.11 + 10.2i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (5.31 + 5.31i)T + 31iT^{2} \) |
| 37 | \( 1 + (7.20 + 0.709i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.589 - 2.96i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-8.85 - 4.73i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (0.717 + 1.73i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (3.52 + 11.6i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-0.840 + 0.689i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (0.616 + 1.15i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-1.60 - 2.99i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (2.75 + 4.11i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-8.18 - 5.47i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (4.55 - 11.0i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-6.13 + 0.603i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (6.10 + 1.21i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (7.50 + 7.50i)T + 97iT^{2} \) |
| 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
−11.12572991246958697011204171841, −9.937533125895463044280962793639, −9.188908415246036150492519591858, −8.293975244572775437816515558316, −6.53213023777295603681694847669, −5.76176153412436785529743858776, −4.55008166588074788813605545477, −3.87221179893431409602513988582, −2.27088053739705820459341620719, −0.15420595179714019161527751578,
3.12798133553684995377998490846, 3.84126500297949866010941610610, 5.36949736430790035251982215780, 6.27877959629697563622260131784, 7.07851392183168458969221612256, 7.81665886648192363226735934435, 8.817505980295825269868193266897, 10.60376547535849415301570332059, 10.72246242587766561297281488733, 12.19971923250177693330010875885
