L(s) = 1 | + (−0.826 + 0.563i)2-s + (1.57 + 0.715i)3-s + (0.365 − 0.930i)4-s + (−0.148 − 0.650i)5-s + (−1.70 + 0.297i)6-s + (2.62 + 0.362i)7-s + (0.222 + 0.974i)8-s + (1.97 + 2.25i)9-s + (0.489 + 0.454i)10-s + (−2.04 − 0.983i)11-s + (1.24 − 1.20i)12-s + (−5.34 + 3.64i)13-s + (−2.36 + 1.17i)14-s + (0.231 − 1.13i)15-s + (−0.733 − 0.680i)16-s + (1.98 + 5.05i)17-s + ⋯ |
L(s) = 1 | + (−0.584 + 0.398i)2-s + (0.910 + 0.413i)3-s + (0.182 − 0.465i)4-s + (−0.0664 − 0.291i)5-s + (−0.696 + 0.121i)6-s + (0.990 + 0.136i)7-s + (0.0786 + 0.344i)8-s + (0.658 + 0.752i)9-s + (0.154 + 0.143i)10-s + (−0.615 − 0.296i)11-s + (0.358 − 0.348i)12-s + (−1.48 + 1.00i)13-s + (−0.633 + 0.314i)14-s + (0.0597 − 0.292i)15-s + (−0.183 − 0.170i)16-s + (0.480 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24130 + 1.07899i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24130 + 1.07899i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 3 | \( 1 + (-1.57 - 0.715i)T \) |
| 7 | \( 1 + (-2.62 - 0.362i)T \) |
good | 5 | \( 1 + (0.148 + 0.650i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (2.04 + 0.983i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (5.34 - 3.64i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.98 - 5.05i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.77 - 6.53i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.24 + 6.57i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-3.30 - 0.498i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.29 - 0.194i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-6.92 - 2.13i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-3.50 + 1.08i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (-7.65 + 5.21i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (5.18 - 0.781i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (7.18 - 2.21i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (0.844 + 2.15i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.29 + 9.14i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.555 + 7.41i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (0.780 - 1.35i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.747 + 0.509i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (4.63 + 3.15i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (6.54 + 11.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.27698848135304942938949480638, −9.374130379198244591400483175218, −8.459255616941090433494898098486, −8.031255122622026662406881441109, −7.33591566508990697887239973854, −6.00778825227481060303308378358, −4.90514151668607605531115556361, −4.21031347879916463182862416815, −2.63854353079114732667795767432, −1.60930638367762949399502893316,
0.917748755042888742833799342632, 2.49123315753624205433547988871, 2.92127034094617520406327274012, 4.47231051252857399244108092925, 5.42174853546864902896683193795, 7.21379743686547015167049977241, 7.50481691430786500631290641070, 8.056547029778910596085450239301, 9.320308467107111753074212181650, 9.699834678869265670883875087822