| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.755 + 0.654i)3-s + (0.841 + 0.540i)4-s + (−0.286 + 1.99i)5-s + (−0.540 − 0.841i)6-s + (−1.39 + 2.24i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (0.836 − 1.83i)10-s + (1.11 + 3.78i)11-s + (0.281 + 0.959i)12-s + (−4.06 − 1.85i)13-s + (1.97 − 1.76i)14-s + (−1.52 + 1.31i)15-s + (0.415 + 0.909i)16-s + (−2.72 + 1.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.436 + 0.378i)3-s + (0.420 + 0.270i)4-s + (−0.128 + 0.891i)5-s + (−0.220 − 0.343i)6-s + (−0.527 + 0.849i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (0.264 − 0.579i)10-s + (0.334 + 1.14i)11-s + (0.0813 + 0.276i)12-s + (−1.12 − 0.514i)13-s + (0.527 − 0.471i)14-s + (−0.393 + 0.340i)15-s + (0.103 + 0.227i)16-s + (−0.660 + 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0339433 + 0.610463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0339433 + 0.610463i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (1.39 - 2.24i)T \) |
| 23 | \( 1 + (0.676 + 4.74i)T \) |
| good | 5 | \( 1 + (0.286 - 1.99i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 3.78i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (4.06 + 1.85i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.72 - 1.74i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.13 + 2.01i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.74 + 3.69i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.313 - 0.271i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (7.16 - 1.02i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.634 - 0.0912i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.60 + 6.59i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 7.63iT - 47T^{2} \) |
| 53 | \( 1 + (-7.22 + 3.29i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.43 - 1.56i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.85 - 2.14i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.97 - 10.1i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.58 - 2.22i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-5.14 + 8.01i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.44 + 0.660i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.973 - 6.76i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (11.1 - 12.8i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (2.20 - 15.3i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.24031549385119871610794705466, −9.709373881827261130957263854447, −8.819340637694572423736011859246, −8.137519667933554043202735915176, −6.91713849286203748074842356027, −6.63575847012247684933885082499, −5.13213256941791499394784207800, −4.01868065217136040764056547562, −2.70893409304322359889688018750, −2.28131859699027148081970911164,
0.32003114714853740696581755338, 1.59218637339019204710773628374, 3.05855971208688146146774356887, 4.20365186496709542121609018459, 5.31363433915170883879403242335, 6.59330004530093080502931990899, 7.06324718473323554240845211819, 8.148242275307632714191238494462, 8.707331245863388534201092535943, 9.454262238216473967668439832994
