L(s) = 1 | + (−0.777 + 0.629i)2-s + (−0.942 + 1.45i)3-s + (0.207 − 0.978i)4-s + (1.99 − 1.01i)5-s + (−0.181 − 1.72i)6-s + (−1.72 − 0.461i)7-s + (0.453 + 0.891i)8-s + (−1.22 − 2.73i)9-s + (−0.912 + 2.04i)10-s + (−4.41 − 0.464i)11-s + (1.22 + 1.22i)12-s + (−1.01 + 1.25i)13-s + (1.62 − 0.724i)14-s + (−0.409 + 3.85i)15-s + (−0.913 − 0.406i)16-s + (−5.47 + 2.78i)17-s + ⋯ |
L(s) = 1 | + (−0.549 + 0.444i)2-s + (−0.544 + 0.838i)3-s + (0.103 − 0.489i)4-s + (0.891 − 0.452i)5-s + (−0.0742 − 0.703i)6-s + (−0.650 − 0.174i)7-s + (0.160 + 0.315i)8-s + (−0.407 − 0.913i)9-s + (−0.288 + 0.645i)10-s + (−1.33 − 0.139i)11-s + (0.353 + 0.353i)12-s + (−0.281 + 0.347i)13-s + (0.434 − 0.193i)14-s + (−0.105 + 0.994i)15-s + (−0.228 − 0.101i)16-s + (−1.32 + 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00307292 - 0.00695693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00307292 - 0.00695693i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.777 - 0.629i)T \) |
| 3 | \( 1 + (0.942 - 1.45i)T \) |
| 5 | \( 1 + (-1.99 + 1.01i)T \) |
good | 7 | \( 1 + (1.72 + 0.461i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.41 + 0.464i)T + (10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.01 - 1.25i)T + (-2.70 - 12.7i)T^{2} \) |
| 17 | \( 1 + (5.47 - 2.78i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (3.58 + 1.16i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.64 - 4.29i)T + (-17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (3.83 - 4.25i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (4.81 + 5.35i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.49 + 9.43i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-10.2 + 1.07i)T + (40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (1.25 - 4.67i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (4.88 - 0.255i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (0.446 + 0.227i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.344 - 3.28i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.05 + 10.0i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-11.1 - 0.582i)T + (66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (3.66 - 1.19i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.95 - 12.3i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.600 - 0.540i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (4.97 - 7.65i)T + (-33.7 - 75.8i)T^{2} \) |
| 89 | \( 1 + (-7.69 - 5.58i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.188 - 3.59i)T + (-96.4 + 10.1i)T^{2} \) |
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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
−11.08614535551434431750397705178, −10.73805435810602417912370339669, −9.614020603007759801395930586493, −9.311052722042557008287144063518, −8.215093643564725402556724142802, −6.86717397578232355289275666969, −5.95168281329145360573576086760, −5.25716996540143332036794095523, −4.08387636696766475732666537439, −2.28211955622855599915546630341,
0.00541780292562631680487768290, 2.10660346370516398362017725667, 2.79713655105799944782770305616, 4.87626341660696770425089302475, 6.03269943984225385060672189486, 6.78026310794314572185521613897, 7.72283650601046584533548375511, 8.755925909062050496440217776192, 9.845099116625675892255690193289, 10.57608141481895263530443045298