L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.0744 − 1.73i)3-s + (0.866 − 0.499i)4-s + (1.68 + 1.47i)5-s + (−0.519 − 1.65i)6-s + (2.42 + 1.05i)7-s + (0.707 − 0.707i)8-s + (−2.98 + 0.257i)9-s + (2.00 + 0.989i)10-s + (−2.50 + 1.44i)11-s + (−0.929 − 1.46i)12-s + (−4.54 − 4.54i)13-s + (2.61 + 0.393i)14-s + (2.42 − 3.01i)15-s + (0.500 − 0.866i)16-s + (0.372 − 1.39i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.0429 − 0.999i)3-s + (0.433 − 0.249i)4-s + (0.751 + 0.659i)5-s + (−0.212 − 0.674i)6-s + (0.916 + 0.399i)7-s + (0.249 − 0.249i)8-s + (−0.996 + 0.0858i)9-s + (0.634 + 0.313i)10-s + (−0.754 + 0.435i)11-s + (−0.268 − 0.421i)12-s + (−1.26 − 1.26i)13-s + (0.699 + 0.105i)14-s + (0.626 − 0.779i)15-s + (0.125 − 0.216i)16-s + (0.0904 − 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73035 - 0.724954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73035 - 0.724954i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.0744 + 1.73i)T \) |
| 5 | \( 1 + (-1.68 - 1.47i)T \) |
| 7 | \( 1 + (-2.42 - 1.05i)T \) |
good | 11 | \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.54 + 4.54i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.372 + 1.39i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.42 - 0.820i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.27 - 4.76i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + (-0.155 - 0.269i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.756 - 2.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 8.54iT - 41T^{2} \) |
| 43 | \( 1 + (-6.09 - 6.09i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 + 1.51i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.518 + 0.138i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.20 + 5.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.348 + 0.603i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.807 - 0.216i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 2.72iT - 71T^{2} \) |
| 73 | \( 1 + (0.0483 - 0.180i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.11 + 4.68i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.26 + 2.26i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.60 - 6.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.75 + 4.75i)T - 97iT^{2} \) |
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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
−12.43743778427099496022824030642, −11.44038731217673168666057202243, −10.59425196056669468477826259200, −9.433821136823509740620767725864, −7.71998960296676014868595036477, −7.29908613101691874726425120184, −5.68420415663166061851137247397, −5.25178205155545780339668794747, −2.93171190484942800659232472414, −1.96789334743234967911382330016,
2.34621396470250715324734986377, 4.21165339734964043864699220925, 4.96338582959582872125956536008, 5.82355459305078981055200090775, 7.40476131856536143330010731916, 8.645005852741994029596121425231, 9.590768328662679540765954511621, 10.65017844801048947113310960705, 11.47987507036034024978811197807, 12.54496163440312769389954996373