| L(s) = 1 | + (−0.279 − 1.38i)2-s + (1.40 − 1.00i)3-s + (−1.84 + 0.774i)4-s + (2.25 + 0.821i)5-s + (−1.79 − 1.67i)6-s + (1.95 + 0.344i)7-s + (1.58 + 2.34i)8-s + (0.965 − 2.84i)9-s + (0.508 − 3.35i)10-s + (0.553 + 1.52i)11-s + (−1.81 + 2.95i)12-s + (−3.57 − 4.26i)13-s + (−0.0681 − 2.80i)14-s + (4.00 − 1.11i)15-s + (2.80 − 2.85i)16-s + (−6.40 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.197 − 0.980i)2-s + (0.812 − 0.582i)3-s + (−0.921 + 0.387i)4-s + (1.00 + 0.367i)5-s + (−0.731 − 0.681i)6-s + (0.738 + 0.130i)7-s + (0.561 + 0.827i)8-s + (0.321 − 0.946i)9-s + (0.160 − 1.06i)10-s + (0.166 + 0.458i)11-s + (−0.524 + 0.851i)12-s + (−0.992 − 1.18i)13-s + (−0.0182 − 0.749i)14-s + (1.03 − 0.288i)15-s + (0.700 − 0.714i)16-s + (−1.55 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13495 - 0.992084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13495 - 0.992084i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.279 + 1.38i)T \) |
| 3 | \( 1 + (-1.40 + 1.00i)T \) |
| good | 5 | \( 1 + (-2.25 - 0.821i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.95 - 0.344i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.553 - 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.57 + 4.26i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.40 - 3.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.465 - 2.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.138 - 0.116i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.76 - 0.311i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.55 + 2.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.85 + 3.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-11.1 + 4.04i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.08 - 11.8i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + (1.96 - 5.40i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (7.60 + 1.34i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 2.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.77 - 8.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.718 + 0.855i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.87 - 5.70i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.25 - 1.91i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−12.34158631610606028556378083845, −11.01054163436078634509991832023, −10.12917126738604349112521715565, −9.270091220365439565559263905303, −8.313322670847610248170059870292, −7.33924486652929175622606819441, −5.78685137726912886738975970224, −4.25674768822615757307370357678, −2.62563727909396636533957900898, −1.80440765013621133790347102046,
2.15325820908301983026435715940, 4.41927134840155852466996782958, 5.00954723535634180047916289091, 6.51273799763741907771473190407, 7.58308183395994038212777518463, 8.932031660814821873196235298459, 9.126369409711437266857405598865, 10.18972378710008234911069130981, 11.37272170086501735016136673166, 13.13831799433321404024037150168
