| L(s) = 1 | + (1.89 − 0.284i)2-s + (1.05 − 1.37i)3-s + (1.58 − 0.487i)4-s + (−0.509 − 0.347i)5-s + (1.59 − 2.90i)6-s + (2.55 − 0.698i)7-s + (−0.595 + 0.286i)8-s + (−0.790 − 2.89i)9-s + (−1.06 − 0.511i)10-s + (1.83 − 0.277i)11-s + (0.989 − 2.68i)12-s + (−1.35 + 3.44i)13-s + (4.62 − 2.04i)14-s + (−1.01 + 0.336i)15-s + (−3.77 + 2.57i)16-s + (0.108 + 0.476i)17-s + ⋯ |
| L(s) = 1 | + (1.33 − 0.201i)2-s + (0.606 − 0.794i)3-s + (0.790 − 0.243i)4-s + (−0.227 − 0.155i)5-s + (0.650 − 1.18i)6-s + (0.964 − 0.263i)7-s + (−0.210 + 0.101i)8-s + (−0.263 − 0.964i)9-s + (−0.335 − 0.161i)10-s + (0.554 − 0.0835i)11-s + (0.285 − 0.776i)12-s + (−0.375 + 0.956i)13-s + (1.23 − 0.547i)14-s + (−0.261 + 0.0868i)15-s + (−0.944 + 0.643i)16-s + (0.0263 + 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.75287 - 1.57712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75287 - 1.57712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.05 + 1.37i)T \) |
| 7 | \( 1 + (-2.55 + 0.698i)T \) |
| good | 2 | \( 1 + (-1.89 + 0.284i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (0.509 + 0.347i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 0.277i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (1.35 - 3.44i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-0.108 - 0.476i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 + (-2.49 + 0.770i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-9.29 - 2.86i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-3.05 - 5.28i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.66 - 7.29i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (7.15 + 4.87i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (5.57 - 3.80i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (0.579 - 0.0872i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.57 + 6.91i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.915 - 12.2i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (10.4 + 3.23i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 5.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.553 - 2.42i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.39 - 1.75i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (0.767 - 1.32i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.22 + 5.67i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.10 + 6.40i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-3.74 + 6.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.61728870201664510910898078992, −10.28753109976449922168448656937, −8.717533237608990416100635333791, −8.402826365528390908541775449970, −6.93891051485509171571850551222, −6.35096880799129856623251315997, −4.83627580853060722074127093459, −4.19048372279343481530052383581, −2.91292743793275486334275775391, −1.66272178645795322483317364587,
2.43765065669704424474817513735, 3.54528209397526034152205147527, 4.53004496186981723969518100853, 5.15335220000342769465593629621, 6.26918563428556128798422628365, 7.61121060176291239299350614491, 8.480684511990823256597504041243, 9.453830863782377268747809559668, 10.52993433317813532517797026369, 11.42917982059303646148186147121
