L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.41 + i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.70 + 0.292i)6-s − 4·7-s + (0.707 − 0.707i)8-s + (1.00 − 2.82i)9-s − 1.00·10-s + 1.41·11-s + (−1.00 − 1.41i)12-s + (−3 − 3i)13-s + (2.82 + 2.82i)14-s + (−0.292 + 1.70i)15-s − 1.00·16-s + (1.41 − 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.816 + 0.577i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.696 + 0.119i)6-s − 1.51·7-s + (0.250 − 0.250i)8-s + (0.333 − 0.942i)9-s − 0.316·10-s + 0.426·11-s + (−0.288 − 0.408i)12-s + (−0.832 − 0.832i)13-s + (0.755 + 0.755i)14-s + (−0.0756 + 0.440i)15-s − 0.250·16-s + (0.342 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4282295428\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4282295428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (6 - i)T \) |
good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4 - 4i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.41 + 1.41i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.82 - 2.82i)T + 29iT^{2} \) |
| 31 | \( 1 + (3 - 3i)T - 31iT^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 + (8.48 - 8.48i)T - 59iT^{2} \) |
| 61 | \( 1 + (6 - 6i)T - 61iT^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + (-5 - 5i)T + 79iT^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (1.41 + 1.41i)T + 89iT^{2} \) |
| 97 | \( 1 + (-6 - 6i)T + 97iT^{2} \) |
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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
−10.13145838804810517787459047232, −9.465627601281878979723435432756, −8.817911709240217777892881106935, −7.47131193638748382339043718047, −6.66638591006437391324509327225, −5.75289344278902563946886756902, −4.93523821512521970069000322291, −3.63805441140259002068395332658, −2.95417104135067089236185922589, −1.07598828802837731205861724513,
0.29347935697828269337349601110, 1.89721627541678866850102460989, 3.23517271497079398645015930388, 4.73010517988797683512350664556, 5.72343930962737497177682735936, 6.49032805634132394371860172616, 6.97239273024662659244090820555, 7.65647354005068150731891413152, 9.051742017795322477606519952128, 9.611315291123783975646318418060