| L(s) = 1 | + (−0.631 + 1.26i)2-s + (0.678 + 0.883i)3-s + (−1.20 − 1.59i)4-s + (−2.77 − 2.13i)5-s + (−1.54 + 0.299i)6-s + (−2.64 + 0.168i)7-s + (2.78 − 0.509i)8-s + (0.455 − 1.69i)9-s + (4.45 − 2.16i)10-s + (−5.46 − 0.719i)11-s + (0.598 − 2.14i)12-s + (−1.04 − 0.433i)13-s + (1.45 − 3.44i)14-s − 3.90i·15-s + (−1.11 + 3.84i)16-s + (−0.0754 − 0.0435i)17-s + ⋯ |
| L(s) = 1 | + (−0.446 + 0.894i)2-s + (0.391 + 0.510i)3-s + (−0.600 − 0.799i)4-s + (−1.24 − 0.953i)5-s + (−0.631 + 0.122i)6-s + (−0.997 + 0.0635i)7-s + (0.983 − 0.180i)8-s + (0.151 − 0.566i)9-s + (1.40 − 0.685i)10-s + (−1.64 − 0.217i)11-s + (0.172 − 0.619i)12-s + (−0.290 − 0.120i)13-s + (0.389 − 0.921i)14-s − 1.00i·15-s + (−0.278 + 0.960i)16-s + (−0.0182 − 0.0105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147601 - 0.172602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147601 - 0.172602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.631 - 1.26i)T \) |
| 7 | \( 1 + (2.64 - 0.168i)T \) |
| good | 3 | \( 1 + (-0.678 - 0.883i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (2.77 + 2.13i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (5.46 + 0.719i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (1.04 + 0.433i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (0.0754 + 0.0435i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.824 - 6.26i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.08 + 4.03i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.818 + 1.97i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (1.68 - 2.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.85 + 5.25i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (2.23 + 2.23i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.669 - 1.61i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.87 + 2.23i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.8 + 1.43i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (0.455 - 3.45i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.17 - 0.154i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (3.42 + 4.46i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-2.00 + 2.00i)T - 71iT^{2} \) |
| 73 | \( 1 + (-6.66 + 1.78i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.42 - 2.55i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.9 + 4.93i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (16.4 + 4.42i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−12.33837079833370443075991946124, −10.61233635918822819526064063251, −9.847414167445254618089529113592, −8.807286391701549784081480336711, −8.145830736708322834181732387629, −7.19462262151065374206372546255, −5.75366197105881606472797294644, −4.60804429315396334087863692905, −3.45924417255286030878437193344, −0.19956190220232295851137265678,
2.58087407496097853127888985982, 3.28470764417457652566481939596, 4.83351783305882289115864940696, 7.11867458569315742786144097796, 7.49362747620915645535817368202, 8.501650973711414894370163657662, 9.835331910600691816104029371611, 10.69886983805569340502776847561, 11.37997441087600163295725673614, 12.54832503773785118560559991879
