| L(s) = 1 | + (1.47 + 1.20i)2-s + (−0.624 − 0.0504i)3-s + (0.325 + 1.59i)4-s + (−0.992 + 0.120i)5-s + (−0.861 − 0.827i)6-s + (−0.770 − 1.21i)7-s + (0.330 − 0.629i)8-s + (−2.57 − 0.418i)9-s + (−1.60 − 1.01i)10-s + (−0.514 − 3.16i)11-s + (−0.122 − 1.01i)12-s + (3.18 − 1.68i)13-s + (0.331 − 2.72i)14-s + (0.626 − 0.0252i)15-s + (4.23 − 1.80i)16-s + (5.28 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (1.04 + 0.851i)2-s + (−0.360 − 0.0291i)3-s + (0.162 + 0.796i)4-s + (−0.443 + 0.0539i)5-s + (−0.351 − 0.337i)6-s + (−0.291 − 0.460i)7-s + (0.116 − 0.222i)8-s + (−0.857 − 0.139i)9-s + (−0.508 − 0.321i)10-s + (−0.155 − 0.954i)11-s + (−0.0354 − 0.292i)12-s + (0.883 − 0.468i)13-s + (0.0884 − 0.728i)14-s + (0.161 − 0.00651i)15-s + (1.05 − 0.451i)16-s + (1.28 − 0.810i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76168 - 0.481959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76168 - 0.481959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.992 - 0.120i)T \) |
| 13 | \( 1 + (-3.18 + 1.68i)T \) |
| good | 2 | \( 1 + (-1.47 - 1.20i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (0.624 + 0.0504i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (0.770 + 1.21i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (0.514 + 3.16i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (-5.28 + 3.34i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (3.88 + 2.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 2.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.14 - 1.39i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.18 - 4.80i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (8.32 - 2.41i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (-0.926 + 11.4i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (-2.71 + 9.37i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (4.93 - 5.56i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (3.28 + 1.72i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (3.07 - 7.21i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-0.179 + 4.46i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (-1.47 - 0.302i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (-10.0 + 4.76i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (2.19 + 0.832i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (-5.30 - 4.70i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-0.0508 + 0.0350i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (5.19 - 3.00i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.62 + 7.48i)T + (-26.9 - 93.1i)T^{2} \) |
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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
−10.44289216390032072047867346682, −9.038130652842770998746884012701, −8.217767274987953776779061565997, −7.24748648525652531441308518337, −6.51836221665163439871509706577, −5.59440217327917060258780231385, −5.14050937836328760722457775253, −3.67706916371292372846347527363, −3.24169314327612396280096904950, −0.68229323182055085374001945669,
1.73542225529677469301003259841, 2.93131481642235477760809921876, 3.86154761823474079990499490193, 4.72739216834138137987539628827, 5.71375655613810617670038304799, 6.37773646007069140192013738451, 7.87727123076775882503002080424, 8.479314288845314830796709446893, 9.696475244811409175960882631636, 10.64703955053531656571484754350
