L(s) = 1 | + (−0.918 − 1.59i)2-s + (−1.56 − 2.70i)3-s + (−0.688 + 1.19i)4-s + (1.63 + 2.83i)5-s + (−2.87 + 4.97i)6-s − 3.63·7-s − 1.14·8-s + (−3.38 + 5.87i)9-s + (3.00 − 5.20i)10-s − 11-s + 4.30·12-s + (0.224 − 0.388i)13-s + (3.33 + 5.77i)14-s + (5.11 − 8.85i)15-s + (2.42 + 4.20i)16-s + (−2.11 − 3.66i)17-s + ⋯ |
L(s) = 1 | + (−0.649 − 1.12i)2-s + (−0.902 − 1.56i)3-s + (−0.344 + 0.595i)4-s + (0.731 + 1.26i)5-s + (−1.17 + 2.03i)6-s − 1.37·7-s − 0.405·8-s + (−1.12 + 1.95i)9-s + (0.949 − 1.64i)10-s − 0.301·11-s + 1.24·12-s + (0.0621 − 0.107i)13-s + (0.891 + 1.54i)14-s + (1.32 − 2.28i)15-s + (0.607 + 1.05i)16-s + (−0.512 − 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.105934 + 0.0938910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105934 + 0.0938910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 19 | \( 1 + (3.96 + 1.82i)T \) |
good | 2 | \( 1 + (0.918 + 1.59i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.56 + 2.70i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.63 - 2.83i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 13 | \( 1 + (-0.224 + 0.388i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.11 + 3.66i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.506 + 0.877i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.37 + 5.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + (-0.641 - 1.11i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.08 - 3.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.38 + 5.85i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.19 - 2.06i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.75 - 8.23i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.33 - 4.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.718 + 1.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.28 + 9.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.32 + 2.29i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.230 - 0.399i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.09T + 83T^{2} \) |
| 89 | \( 1 + (-2.31 + 4.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.62 + 8.01i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.53781679538705677767299210769, −10.71510828655367504241868430420, −10.11201559061454038498668570335, −8.896428051201244487045764871720, −7.22769160759113462316127147029, −6.52856633379932823753212627282, −5.83621902594906652700621969632, −2.93426154206872820183079901465, −2.13276446198460661754991724304, −0.15212101577789393005132223629,
3.70402230011527282079545244075, 5.12604767183538157008783791667, 5.86381448876870529883472179922, 6.67737398214200015986501890388, 8.661989602240053045730393880133, 9.105687346626292575085285555703, 9.928781232225153702679992347634, 10.70152489036232728213168826346, 12.30823223822713869546808292250, 12.88425339988613467635774702845