| L(s) = 1 | + (−0.797 − 1.16i)2-s + (−0.729 + 1.86i)4-s + (2.65 − 0.467i)5-s + (0.448 − 0.534i)7-s + (2.75 − 0.632i)8-s + (−2.66 − 2.72i)10-s + (−1.10 + 6.24i)11-s + (1.64 + 0.598i)13-s + (−0.981 − 0.0978i)14-s + (−2.93 − 2.71i)16-s + (4.12 − 2.38i)17-s + (0.795 + 0.459i)19-s + (−1.06 + 5.27i)20-s + (8.17 − 3.68i)22-s + (4.88 − 4.09i)23-s + ⋯ |
| L(s) = 1 | + (−0.563 − 0.826i)2-s + (−0.364 + 0.931i)4-s + (1.18 − 0.209i)5-s + (0.169 − 0.201i)7-s + (0.974 − 0.223i)8-s + (−0.841 − 0.861i)10-s + (−0.331 + 1.88i)11-s + (0.455 + 0.165i)13-s + (−0.262 − 0.0261i)14-s + (−0.733 − 0.679i)16-s + (1.00 − 0.578i)17-s + (0.182 + 0.105i)19-s + (−0.237 + 1.18i)20-s + (1.74 − 0.786i)22-s + (1.01 − 0.853i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14554 - 0.394906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14554 - 0.394906i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.797 + 1.16i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.65 + 0.467i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.448 + 0.534i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (1.10 - 6.24i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.64 - 0.598i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.12 + 2.38i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.795 - 0.459i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.88 + 4.09i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.69 + 4.66i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.35 + 1.61i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.943 + 1.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.208 - 0.571i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.489 + 0.0863i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.04 + 4.23i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 5.84iT - 53T^{2} \) |
| 59 | \( 1 + (0.785 + 4.45i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.35 - 2.81i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.703 + 1.93i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.99 - 6.91i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.46 - 9.47i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.23 + 14.3i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (11.0 - 4.01i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (8.08 + 4.67i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0861 + 0.488i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.43902420369074420792422162779, −10.30437845038958258384766751762, −9.797438107999773537232580299107, −9.087053297391476725584819178464, −7.81862154653814685193041422235, −6.91925132648273239272984405444, −5.36118827480019469716392884180, −4.30918615404192833565500151199, −2.62226243819966579703472930409, −1.51028791673472981716046865671,
1.37137586688872188160863832840, 3.26510045314367936473009089604, 5.40944763328487892693743925329, 5.72695330160762391594331626490, 6.78678492173655553398229057154, 8.078930033319973681206580299799, 8.799440170137772379857542488231, 9.715102360630633318105809624546, 10.62243258620362025635750450975, 11.30402217533178432829254986740
