| L(s) = 1 | + (−0.775 − 1.18i)2-s + (−0.162 + 1.72i)3-s + (−0.795 + 1.83i)4-s + (2.44 + 1.71i)5-s + (2.16 − 1.14i)6-s + (2.32 + 1.95i)7-s + (2.78 − 0.483i)8-s + (−2.94 − 0.559i)9-s + (0.126 − 4.21i)10-s + (−3.36 + 2.35i)11-s + (−3.03 − 1.66i)12-s + (3.58 − 1.67i)13-s + (0.503 − 4.26i)14-s + (−3.34 + 3.93i)15-s + (−2.73 − 2.92i)16-s + (−1.29 + 0.745i)17-s + ⋯ |
| L(s) = 1 | + (−0.548 − 0.836i)2-s + (−0.0937 + 0.995i)3-s + (−0.397 + 0.917i)4-s + (1.09 + 0.764i)5-s + (0.883 − 0.467i)6-s + (0.879 + 0.738i)7-s + (0.985 − 0.170i)8-s + (−0.982 − 0.186i)9-s + (0.0400 − 1.33i)10-s + (−1.01 + 0.711i)11-s + (−0.876 − 0.482i)12-s + (0.993 − 0.463i)13-s + (0.134 − 1.14i)14-s + (−0.863 + 1.01i)15-s + (−0.683 − 0.730i)16-s + (−0.313 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.944037 + 0.675203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.944037 + 0.675203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.775 + 1.18i)T \) |
| 3 | \( 1 + (0.162 - 1.72i)T \) |
| good | 5 | \( 1 + (-2.44 - 1.71i)T + (1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 1.95i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (3.36 - 2.35i)T + (3.76 - 10.3i)T^{2} \) |
| 13 | \( 1 + (-3.58 + 1.67i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (1.29 - 0.745i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.53 - 0.679i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (4.76 + 5.67i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.74 - 2.21i)T + (18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (-6.48 - 7.72i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.325 + 1.21i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (6.38 + 2.32i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.88 - 5.55i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-0.0868 - 0.0729i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.80 + 5.80i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.39 + 7.70i)T + (-20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (-2.42 + 0.212i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-1.06 - 2.28i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (14.1 - 8.14i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.0580 - 0.0335i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.351 - 0.965i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.674 - 1.44i)T + (-53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (-0.0821 + 0.142i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.126 - 0.715i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.94115290035489502021477600968, −10.31841689743313937499093027970, −10.00341053312789774415073832661, −8.591651409710451435533420568525, −8.325131109852343085903104777802, −6.55777892606849208573646901861, −5.40161389284916728654915021723, −4.42923353899882807030849549966, −2.91973578324718914160995642869, −2.08758182715094116982982493274,
0.941923026934429688848603693966, 2.03192209064045506110882990781, 4.52806483028494258559454077682, 5.67570716494896782108112889005, 6.16379082655263576098259050847, 7.40132344279462296723740605892, 8.242468804037859891276597244062, 8.755569785117882636627656905212, 9.969053371398767232947196075457, 10.85658602517379788611165060184
