L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (−1.82 + 3.15i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.44 + 0.834i)11-s + 0.999i·12-s + (2.82 − 2.24i)13-s + 3.64·14-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s − 0.999·18-s + (−5.46 − 3.15i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.688 + 1.19i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.435 + 0.251i)11-s + 0.288i·12-s + (0.782 − 0.622i)13-s + 0.974·14-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s − 0.235·18-s + (−1.25 − 0.724i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1169610275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1169610275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.82 + 2.24i)T \) |
good | 7 | \( 1 + (1.82 - 3.15i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.44 - 0.834i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.46 + 3.15i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 - 0.622i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.02 - 8.69i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.21iT - 31T^{2} \) |
| 37 | \( 1 + (4.93 + 8.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.04 + 4.64i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.55 + 3.78i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 + 0.848iT - 53T^{2} \) |
| 59 | \( 1 + (5.29 + 3.05i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.37 + 12.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.04 + 1.75i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.93T + 79T^{2} \) |
| 83 | \( 1 + 7.95T + 83T^{2} \) |
| 89 | \( 1 + (5.15 - 2.97i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.37 - 2.38i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.903969543456045774264879842491, −8.324545327489028137962928543102, −7.21609003344325783865395973319, −6.49261127717890212691225266357, −5.52728031971486595531228457119, −4.51494532947369908653286164664, −3.28936711293253176984669199846, −2.68791264755028670974620170878, −1.76915723045215910315676390544, −0.04243813565378289368779543545,
1.56744906172517292990162975616, 2.95747482854840611284755990354, 4.20600735297484720824903377648, 4.38318976117954656536657176286, 6.08405568411828696440862849222, 6.40461801450315496260044832258, 7.33254539597617156345932606966, 8.292048640702432610505841337983, 8.541584552176728481520110464189, 9.730883765477916494647650977544