| L(s) = 1 | + (−0.939 + 1.05i)2-s + (−0.494 + 1.66i)3-s + (−0.234 − 1.98i)4-s + (−0.0359 − 0.273i)5-s + (−1.29 − 2.08i)6-s + (−1.13 − 4.22i)7-s + (2.32 + 1.61i)8-s + (−2.51 − 1.64i)9-s + (0.322 + 0.218i)10-s + (3.37 − 2.58i)11-s + (3.41 + 0.591i)12-s + (2.42 − 3.15i)13-s + (5.53 + 2.77i)14-s + (0.471 + 0.0752i)15-s + (−3.88 + 0.932i)16-s − 1.15·17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.747i)2-s + (−0.285 + 0.958i)3-s + (−0.117 − 0.993i)4-s + (−0.0160 − 0.122i)5-s + (−0.526 − 0.849i)6-s + (−0.427 − 1.59i)7-s + (0.820 + 0.571i)8-s + (−0.837 − 0.546i)9-s + (0.102 + 0.0691i)10-s + (1.01 − 0.780i)11-s + (0.985 + 0.170i)12-s + (0.671 − 0.875i)13-s + (1.47 + 0.741i)14-s + (0.121 + 0.0194i)15-s + (−0.972 + 0.233i)16-s − 0.279·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687010 - 0.0907168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687010 - 0.0907168i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 1.05i)T \) |
| 3 | \( 1 + (0.494 - 1.66i)T \) |
| good | 5 | \( 1 + (0.0359 + 0.273i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (1.13 + 4.22i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.37 + 2.58i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.42 + 3.15i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 + (7.29 - 3.02i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (3.36 + 0.901i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-8.61 - 1.13i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-0.264 + 0.152i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.12 + 7.53i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.968 - 3.61i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.90 + 3.78i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (4.44 + 2.56i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.95 - 4.72i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.60 + 0.343i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.02 + 7.75i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-4.08 + 5.32i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-1.28 - 1.28i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.601 - 1.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (13.1 + 1.72i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (5.53 - 5.53i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.53 + 4.38i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21825872185468203645834119945, −10.52801970237733227598863581500, −10.05951531715524822292125448462, −8.816144284238359386480877471662, −8.169148607935517006358004247436, −6.62970134074800862353344345046, −6.11209564041562641885710765607, −4.57584584770784984817390420998, −3.68396403166149124623714964220, −0.69604961782079123100653626396,
1.74373675944178985971439032643, 2.74784249264097882037596586805, 4.52480267985614889700697797262, 6.35676625283051331198259549128, 6.79186940666937511970461204303, 8.485063366234962400722202446177, 8.779243608679542426193627149918, 9.910322523588690282663742751324, 11.25916877421763273354850799080, 11.80243864427267729122471726076
