| L(s) = 1 | + (0.142 + 0.989i)3-s + (2.52 + 0.742i)5-s + (−1.72 + 1.99i)7-s + (−0.959 + 0.281i)9-s + (−4.25 + 2.73i)11-s + (1.58 + 1.83i)13-s + (−0.374 + 2.60i)15-s + (−0.390 + 0.854i)17-s + (−0.296 − 0.649i)19-s + (−2.22 − 1.42i)21-s + (4.51 − 1.60i)23-s + (1.63 + 1.04i)25-s + (−0.415 − 0.909i)27-s + (0.132 − 0.290i)29-s + (−0.0905 + 0.629i)31-s + ⋯ |
| L(s) = 1 | + (0.0821 + 0.571i)3-s + (1.13 + 0.331i)5-s + (−0.653 + 0.754i)7-s + (−0.319 + 0.0939i)9-s + (−1.28 + 0.823i)11-s + (0.440 + 0.507i)13-s + (−0.0967 + 0.673i)15-s + (−0.0946 + 0.207i)17-s + (−0.0680 − 0.149i)19-s + (−0.484 − 0.311i)21-s + (0.941 − 0.335i)23-s + (0.326 + 0.209i)25-s + (−0.0799 − 0.175i)27-s + (0.0246 − 0.0539i)29-s + (−0.0162 + 0.113i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.898133 + 1.14443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898133 + 1.14443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-4.51 + 1.60i)T \) |
| good | 5 | \( 1 + (-2.52 - 0.742i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (1.72 - 1.99i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.25 - 2.73i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.58 - 1.83i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.390 - 0.854i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.296 + 0.649i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.132 + 0.290i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0905 - 0.629i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (4.11 - 1.20i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-9.99 - 2.93i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0453 - 0.315i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 0.490T + 47T^{2} \) |
| 53 | \( 1 + (-2.78 + 3.21i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-9.45 - 10.9i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.854 - 5.94i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.96 + 2.54i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.69 - 4.30i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.28 + 11.5i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (9.35 + 10.7i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-13.2 + 3.88i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.86 + 12.9i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-15.3 - 4.51i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.68473502100735343447390657115, −10.18276667483830232209934400284, −9.349202313841583852753483825533, −8.721190204679763252528240223061, −7.37665855457051043569686436372, −6.28510868187384137767623745644, −5.56418875682237352796044206419, −4.55479534471159815034662801162, −2.98998295917525044272608884920, −2.16873541878295443562397459633,
0.820642940528903979534113655550, 2.41275274552089154065768333628, 3.52434197243944946832211212281, 5.23297084574187406979635513997, 5.87278770623963422148230363597, 6.87650731684180749742919024113, 7.83090878431201329253385746330, 8.772966428290491789348108389819, 9.693341924469667917174314838160, 10.51648181182001067777829563814
