| L(s) = 1 | + (−0.685 − 1.58i)2-s + (0.844 + 1.51i)3-s + (−0.684 + 0.725i)4-s + (0.0800 − 1.37i)5-s + (1.82 − 2.37i)6-s + (3.19 − 0.758i)7-s + (−1.63 − 0.593i)8-s + (−1.57 + 2.55i)9-s + (−2.24 + 0.815i)10-s + (0.171 − 0.112i)11-s + (−1.67 − 0.422i)12-s + (−5.21 + 0.610i)13-s + (−3.39 − 4.56i)14-s + (2.14 − 1.03i)15-s + (0.290 + 4.99i)16-s + (3.88 + 3.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.484 − 1.12i)2-s + (0.487 + 0.873i)3-s + (−0.342 + 0.362i)4-s + (0.0358 − 0.614i)5-s + (0.745 − 0.971i)6-s + (1.20 − 0.286i)7-s + (−0.576 − 0.209i)8-s + (−0.524 + 0.851i)9-s + (−0.708 + 0.257i)10-s + (0.0517 − 0.0340i)11-s + (−0.483 − 0.121i)12-s + (−1.44 + 0.169i)13-s + (−0.908 − 1.22i)14-s + (0.554 − 0.268i)15-s + (0.0726 + 1.24i)16-s + (0.942 + 0.791i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.792303 - 0.446664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792303 - 0.446664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.844 - 1.51i)T \) |
| good | 2 | \( 1 + (0.685 + 1.58i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.0800 + 1.37i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-3.19 + 0.758i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-0.171 + 0.112i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (5.21 - 0.610i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-3.88 - 3.26i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (2.25 - 1.88i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (3.66 + 0.867i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (3.76 - 5.05i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (1.45 + 4.86i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 8.19i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (1.09 - 2.53i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (1.41 + 0.712i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.596 + 1.99i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-4.51 + 7.82i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-12.4 - 8.15i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.03 - 1.09i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (8.00 + 10.7i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-4.39 + 1.59i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.3 - 5.56i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (0.588 + 1.36i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (5.64 + 13.0i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (4.83 + 1.75i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.104 - 1.79i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−14.53458172345081895874484723077, −12.79459415084425983091728082633, −11.77913664534398246618467814628, −10.71934085890177500965271870313, −9.932490482577965094985389341222, −8.881439586641900472400926314392, −7.84983937197080177225573352789, −5.30568391255568530855787326727, −3.95281083076201477292719855829, −2.04419744164932970852929470195,
2.59302461954323681844956235086, 5.33573838181312326347458291348, 6.80805311331549717070023914489, 7.61343386712990279813043126889, 8.399036635831916131515059076427, 9.703341520031661151886421792954, 11.52025562688008663133321903582, 12.32430465451424339492459569538, 14.01616277054094116089620760663, 14.67401762037268771886757008717
