| L(s) = 1 | + (1.36 + 0.366i)2-s + (1.61 + 2.79i)3-s + (1.73 + i)4-s + (0.725 − 0.725i)5-s + (1.18 + 4.41i)6-s + (−9.51 + 2.54i)7-s + (1.99 + 2i)8-s + (−0.720 + 1.24i)9-s + (1.25 − 0.725i)10-s + (−5.57 + 20.8i)11-s + 6.46i·12-s − 13.9·14-s + (3.20 + 0.858i)15-s + (1.99 + 3.46i)16-s + (2.12 + 1.22i)17-s + (−1.44 + 1.44i)18-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.538 + 0.932i)3-s + (0.433 + 0.250i)4-s + (0.145 − 0.145i)5-s + (0.197 + 0.735i)6-s + (−1.35 + 0.364i)7-s + (0.249 + 0.250i)8-s + (−0.0800 + 0.138i)9-s + (0.125 − 0.0725i)10-s + (−0.507 + 1.89i)11-s + 0.538i·12-s − 0.995·14-s + (0.213 + 0.0572i)15-s + (0.124 + 0.216i)16-s + (0.124 + 0.0720i)17-s + (−0.0800 + 0.0800i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27261 + 2.12691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27261 + 2.12691i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + (-1.61 - 2.79i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-0.725 + 0.725i)T - 25iT^{2} \) |
| 7 | \( 1 + (9.51 - 2.54i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (5.57 - 20.8i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (-2.12 - 1.22i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-5.99 - 22.3i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (-21.0 + 12.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (14.4 + 24.9i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-30.8 + 30.8i)T - 961iT^{2} \) |
| 37 | \( 1 + (-12.5 + 46.7i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (25.0 + 6.72i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-5.79 - 3.34i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (9.85 + 9.85i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 36.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-0.269 + 0.0722i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (6.76 - 11.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (42.8 + 11.4i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-12.9 - 48.2i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-62.1 - 62.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 30.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-84.7 + 84.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-18.2 + 67.9i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (18.5 + 69.0i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.91385180801916164627271915916, −10.37285929936204703494071989664, −9.774524745383772423607206618023, −9.163051959802579851134236314790, −7.72318287987087485066333714714, −6.72312787186656900061394492085, −5.58672221931558647666898453188, −4.45799471827637274264262407246, −3.54453727563623965673072195332, −2.39301561192134855676373824469,
0.876123276663128527670873914317, 2.82529364381528550843327239829, 3.27473365261966406608057881979, 5.09081215126088741300283727240, 6.36433673461396597782804135094, 6.89788329283688888529473698619, 8.075277428716937458140082787813, 9.073649235774452250967062872658, 10.30067608414580576611693777400, 11.08457004709405610975607953424
