| L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (2.16 − 0.635i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (1.47 − 1.70i)10-s + (−5.26 − 3.38i)11-s + (0.841 + 0.540i)12-s + (3.13 − 3.61i)13-s + (−0.959 − 0.281i)14-s + (0.320 + 2.23i)15-s + (−0.654 − 0.755i)16-s + (−0.662 − 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (0.967 − 0.284i)5-s + (0.169 + 0.371i)6-s + (−0.247 − 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.467 − 0.538i)10-s + (−1.58 − 1.02i)11-s + (0.242 + 0.156i)12-s + (0.869 − 1.00i)13-s + (−0.256 − 0.0752i)14-s + (0.0828 + 0.576i)15-s + (−0.163 − 0.188i)16-s + (−0.160 − 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70029 - 1.41000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70029 - 1.41000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-1.62 + 4.51i)T \) |
| good | 5 | \( 1 + (-2.16 + 0.635i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (5.26 + 3.38i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.13 + 3.61i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.662 + 1.45i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 3.66i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.50 - 7.66i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.768 - 5.34i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.64 - 0.481i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 0.826i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.800 + 5.57i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 0.112T + 47T^{2} \) |
| 53 | \( 1 + (-7.99 - 9.23i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.22 - 7.17i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.462 - 3.21i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.54 + 3.56i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.46 + 3.51i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.296 - 0.648i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.01 - 2.32i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (16.7 + 4.92i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.707 + 4.92i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-13.6 + 4.00i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.22119275250095388496618406278, −9.058328686627473724738536891511, −8.444198454280271721386382029058, −7.14661288002269181998753398515, −5.95287188308777393127334904942, −5.44494005296164957048708910389, −4.69376404547390339658060646987, −3.25231947923282280546665881818, −2.68520450894204079539134985811, −0.848767797688178484555230390380,
1.88149796167686052147087067615, 2.62883791873238579978229728626, 4.04228988101397789371666150948, 5.23634308381129917033339256726, 5.97486128760127651968286332938, 6.57703910874509920957952088510, 7.60527283359247580840195868442, 8.234235279920635986345557212481, 9.525978613181469087592307686050, 10.06282858258723564369458691385
