L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s + (−0.866 + 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s − 0.517·19-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s + (−0.866 + 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s − 0.517·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6481375097\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6481375097\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 0.517T + T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−8.432435944427906765158974147700, −8.106612534769451666372839021936, −7.50316437308409556321107170327, −6.98523241149691274013264634919, −5.98193446536187816825486688000, −4.88892235706156789802828303688, −4.07823360161446120792239997221, −2.74910282786929233263824091705, −1.78810241226097208659440402209, −0.49680897305987969284577924738,
1.90103401652176705097782572423, 2.56658458657930024329762554417, 3.67433657278697102811524990280, 4.29848605313513106453533295689, 5.16863432004450179869494035249, 6.61869117822908851581308511769, 7.57092187189531419194607071002, 7.990657664369827026846573407130, 8.557272523689157424180071676692, 9.430257524266972390701374515031