L(s) = 1 | − 2-s − i·3-s + 4-s − i·5-s + i·6-s − 8-s + 2·9-s + i·10-s − 6i·11-s − i·12-s − 5·13-s − 15-s + 16-s + (4 − i)17-s − 2·18-s + 3·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.353·8-s + 0.666·9-s + 0.316i·10-s − 1.80i·11-s − 0.288i·12-s − 1.38·13-s − 0.258·15-s + 0.250·16-s + (0.970 − 0.242i)17-s − 0.471·18-s + 0.688·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.661850 - 0.516757i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.661850 - 0.516757i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 + (-4 + i)T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - iT - 29T^{2} \) |
| 31 | \( 1 - 7iT - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 12iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + 3iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 9iT - 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−12.36265902060199657020855747812, −11.70416152727181662473690338497, −10.41184485445080752737826829294, −9.517224951693734740414588796622, −8.351184073823879274480511207488, −7.55489761825516621496059546104, −6.41214605013478270568276589895, −5.06925577053071545335203107003, −3.05268001774041829351668466912, −1.08657951342704377928838709233,
2.19185027504633142046958447704, 4.03515190005163281835807438076, 5.38363727405958443972670767649, 7.25067159768810349849370774076, 7.47260858679348041521214588364, 9.407551739869589008132607484608, 9.849399957671859471824036832083, 10.60233511927826921436502384283, 12.00723022924032505724382508335, 12.61603016955695792603578429928