| L(s) = 1 | + (−1.13 + 0.841i)2-s + (1.61 + 0.634i)3-s + (0.583 − 1.91i)4-s + (0.335 + 2.55i)5-s + (−2.36 + 0.635i)6-s + (−0.254 − 0.949i)7-s + (0.947 + 2.66i)8-s + (2.19 + 2.04i)9-s + (−2.52 − 2.61i)10-s + (−0.837 + 0.642i)11-s + (2.15 − 2.71i)12-s + (−1.15 + 1.50i)13-s + (1.08 + 0.864i)14-s + (−1.07 + 4.32i)15-s + (−3.31 − 2.23i)16-s + 4.36·17-s + ⋯ |
| L(s) = 1 | + (−0.803 + 0.595i)2-s + (0.930 + 0.366i)3-s + (0.291 − 0.956i)4-s + (0.150 + 1.14i)5-s + (−0.965 + 0.259i)6-s + (−0.0961 − 0.358i)7-s + (0.334 + 0.942i)8-s + (0.731 + 0.681i)9-s + (−0.799 − 0.827i)10-s + (−0.252 + 0.193i)11-s + (0.621 − 0.783i)12-s + (−0.320 + 0.418i)13-s + (0.290 + 0.231i)14-s + (−0.277 + 1.11i)15-s + (−0.829 − 0.557i)16-s + 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.802009 + 0.889342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802009 + 0.889342i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.13 - 0.841i)T \) |
| 3 | \( 1 + (-1.61 - 0.634i)T \) |
| good | 5 | \( 1 + (-0.335 - 2.55i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.254 + 0.949i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.837 - 0.642i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.15 - 1.50i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + (-0.0481 + 0.0199i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.476 - 0.127i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (8.61 + 1.13i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.37 - 8.14i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.939 - 3.50i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (6.85 + 8.92i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-2.36 - 1.36i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.72 + 6.57i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.89 + 0.776i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.240 + 1.83i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-9.22 + 12.0i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (10.6 + 10.6i)T + 71iT^{2} \) |
| 73 | \( 1 + (-6.06 + 6.06i)T - 73iT^{2} \) |
| 79 | \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.23 - 0.952i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (3.67 - 3.67i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.21 + 3.82i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.76447003387694709743690663189, −10.58785530561325871130671535338, −10.08379554888480275602511405194, −9.311781695536435573759059616652, −8.109731997978845819345243195767, −7.36660726089260114129538117906, −6.54994575742374911627032554093, −5.08298698152107599408180210956, −3.45267917674897640354238731505, −2.10439474040036622548020486197,
1.17876038678472943641610076204, 2.60211824018526973381956673369, 3.86200194266691752081133508994, 5.45306192537458904572985105771, 7.13967718553409835351031574218, 8.049480657498212085822195710341, 8.769912258339002363607260286007, 9.473928768947573984636409746705, 10.32616691607511863973847426585, 11.71734490802559787423627659482
