| L(s) = 1 | + (0.984 + 0.173i)2-s + (−1.47 − 0.900i)3-s + (0.939 + 0.342i)4-s + (−1.23 + 1.03i)5-s + (−1.30 − 1.14i)6-s + (1.54 + 2.14i)7-s + (0.866 + 0.5i)8-s + (1.37 + 2.66i)9-s + (−1.39 + 0.805i)10-s + (−0.858 + 1.02i)11-s + (−1.08 − 1.35i)12-s + (3.00 − 0.530i)13-s + (1.15 + 2.38i)14-s + (2.75 − 0.421i)15-s + (0.766 + 0.642i)16-s + (1.40 + 2.43i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−0.854 − 0.519i)3-s + (0.469 + 0.171i)4-s + (−0.551 + 0.463i)5-s + (−0.531 − 0.466i)6-s + (0.585 + 0.810i)7-s + (0.306 + 0.176i)8-s + (0.459 + 0.888i)9-s + (−0.441 + 0.254i)10-s + (−0.258 + 0.308i)11-s + (−0.312 − 0.390i)12-s + (0.834 − 0.147i)13-s + (0.308 + 0.636i)14-s + (0.712 − 0.108i)15-s + (0.191 + 0.160i)16-s + (0.341 + 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37619 + 0.620709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37619 + 0.620709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.47 + 0.900i)T \) |
| 7 | \( 1 + (-1.54 - 2.14i)T \) |
| good | 5 | \( 1 + (1.23 - 1.03i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (0.858 - 1.02i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.00 + 0.530i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.40 - 2.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 1.47i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.215 - 0.591i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.585 + 0.103i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.88 + 5.18i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.27 - 7.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.515 + 2.92i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (8.07 + 6.77i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (7.18 - 2.61i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 9.51iT - 53T^{2} \) |
| 59 | \( 1 + (4.31 - 3.62i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.62 + 4.47i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.70 + 9.64i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.73 + 3.31i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.98 - 4.61i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.490 + 2.78i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.953 + 5.41i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (5.41 - 9.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.0 + 13.2i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.54141624116800090169827795870, −11.06390897745244621988074472364, −9.945105384000268464790703997325, −8.270280043742600211565704452962, −7.65389617429929400326023663516, −6.52059565174989111026836939328, −5.69725368627196754115741263291, −4.79966923606194057350400497073, −3.41286955972874482107624600035, −1.80278057936429505728183909519,
1.01249444521455458243336629754, 3.40362503917120653936326588728, 4.41490813800781094676934079893, 5.10000905728013110797281960339, 6.23481756213765103143663698063, 7.30948873991655029257189711892, 8.375297031025388087585125666365, 9.681980549014437931104919292301, 10.67625942451098803997164490358, 11.32887021315580879572562916956
