L(s) = 1 | − i·2-s + (−1 − i)3-s − 4-s + (−1 + i)6-s + (3 − 3i)7-s + i·8-s − i·9-s + (1 − i)11-s + (1 + i)12-s + 4·13-s + (−3 − 3i)14-s + 16-s + (4 + i)17-s − 18-s + 6i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (−0.408 + 0.408i)6-s + (1.13 − 1.13i)7-s + 0.353i·8-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (0.288 + 0.288i)12-s + 1.10·13-s + (−0.801 − 0.801i)14-s + 0.250·16-s + (0.970 + 0.242i)17-s − 0.235·18-s + 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.466482 - 1.35545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.466482 - 1.35545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (-4 - i)T \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1 + i)T - 11iT^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (-5 + 5i)T - 23iT^{2} \) |
| 29 | \( 1 + (7 + 7i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1 - i)T + 31iT^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - 41iT^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + (-1 - i)T + 71iT^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 + (3 - 3i)T - 79iT^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 5i)T + 97iT^{2} \) |
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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
−10.19055548237088168767172942148, −9.026623248452682743650138811226, −8.109386937530317831445252746055, −7.40766840144040713832850647635, −6.27079766460246362622380238879, −5.47474962551724399214730595059, −4.18242729770468026803990015564, −3.52330273059902051275742391786, −1.62322294808888195778274471583, −0.888754554633380140446061525417,
1.60897026600974170888033744264, 3.32020629862830092142368557456, 4.74229828111069674485961639921, 5.21190057555947992023819003364, 5.90489083813684402066817454463, 7.11679163027298041429583442669, 7.975658830428325545234153986964, 8.888695515151238175130683037177, 9.392783880181080440285637716691, 10.66843554127202155280612612488