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 − 2·7-s + (0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s − 1.00·10-s − 2.82·11-s + (1.00 − 1.41i)12-s + (−3 + 3i)13-s + (1.41 − 1.41i)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 − 0.755·7-s + (0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s − 0.316·10-s − 0.852·11-s + (0.288 − 0.408i)12-s + (−0.832 + 0.832i)13-s + (0.377 − 0.377i)14-s + (0.0756 + 0.440i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8672846867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8672846867\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 6i)T \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + (1 - i)T - 19iT^{2} \) |
| 23 | \( 1 + (5.65 + 5.65i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.41 + 1.41i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1 - i)T + 31iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (2.82 + 2.82i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9 - 9i)T + 61iT^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + (-3 + 3i)T - 79iT^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (-7.07 + 7.07i)T - 89iT^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−10.01817937744946976879595112319, −9.614796113478722431272988127242, −8.532626322373114261609186256585, −8.007844563322058477479786869695, −6.99698298852707931511262404097, −6.25628753844529620627550210921, −5.12934463129489083380854484738, −4.18944925528179943722425542347, −2.94120200242710195469088608689, −2.05120581555014585016485864331,
0.37185346704199956613551381823, 1.94255174828790417541021559545, 2.84015135250746152488642334774, 3.66849998151823436401108988228, 5.10748749137227805131380052097, 6.16153169140429698541958761165, 7.29048148965122566274021782540, 7.82718986756229748530228392172, 8.626739136145718912353280464118, 9.632781547312920711115918691044