| L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.06 − 1.76i)5-s + (−4.16 − 5.62i)7-s + 2.82i·8-s + (−2.50 − 4.33i)10-s + (−0.0940 + 0.0542i)11-s − 23.7·13-s + (−1.12 − 9.83i)14-s + (−2.00 + 3.46i)16-s + (−16.6 + 9.63i)17-s + (3.43 − 5.95i)19-s − 7.07i·20-s − 0.153·22-s + (1.56 + 0.904i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.612 − 0.353i)5-s + (−0.595 − 0.803i)7-s + 0.353i·8-s + (−0.250 − 0.433i)10-s + (−0.00854 + 0.00493i)11-s − 1.82·13-s + (−0.0805 − 0.702i)14-s + (−0.125 + 0.216i)16-s + (−0.981 + 0.566i)17-s + (0.180 − 0.313i)19-s − 0.353i·20-s − 0.00697·22-s + (0.0680 + 0.0393i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.148819 - 0.406984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148819 - 0.406984i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4.16 + 5.62i)T \) |
| good | 5 | \( 1 + (3.06 + 1.76i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (0.0940 - 0.0542i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 23.7T + 169T^{2} \) |
| 17 | \( 1 + (16.6 - 9.63i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.43 + 5.95i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-1.56 - 0.904i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 6.62iT - 841T^{2} \) |
| 31 | \( 1 + (21.0 + 36.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-15.6 + 27.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 6.90iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-68.7 - 39.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-82.4 + 47.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (82.6 - 47.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (21.5 - 37.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (33.3 + 57.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (28.1 + 48.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (15.6 - 27.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 18.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (101. + 58.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 65.0T + 9.40e3T^{2} \) |
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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
−10.90584277969128004856523057345, −9.898841524442476441755565761507, −8.884237788638835031709197896774, −7.59298606008740774561940192050, −7.15776700285636956514624964019, −5.93614502165776990677601961780, −4.63425791407192586455963538826, −3.97590584873330983211520833733, −2.51307657125528166956409381598, −0.14386537421325838742177464466,
2.30338447585084587314168694557, 3.23752747674785066727494337742, 4.56998273543243781181625835860, 5.52741304109676128801512376515, 6.77272605335640444405422430873, 7.51438406080549522347701890943, 8.939404943810547300228128822752, 9.758915327827112022689066369478, 10.72188163325357591037645910791, 11.82828220141720527492618145982
