L(s) = 1 | − 2-s + (−1.68 − 0.396i)3-s + 4-s + (2.18 − 0.469i)5-s + (1.68 + 0.396i)6-s − 8-s + (2.68 + 1.33i)9-s + (−2.18 + 0.469i)10-s − 0.939i·11-s + (−1.68 − 0.396i)12-s + 2·13-s + (−3.87 − 0.0737i)15-s + 16-s + 6.63i·17-s + (−2.68 − 1.33i)18-s + 3.46i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.973 − 0.228i)3-s + 0.5·4-s + (0.977 − 0.210i)5-s + (0.688 + 0.161i)6-s − 0.353·8-s + (0.895 + 0.445i)9-s + (−0.691 + 0.148i)10-s − 0.283i·11-s + (−0.486 − 0.114i)12-s + 0.554·13-s + (−0.999 − 0.0190i)15-s + 0.250·16-s + 1.60i·17-s + (−0.633 − 0.314i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9935642999\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9935642999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.68 + 0.396i)T \) |
| 5 | \( 1 + (-2.18 + 0.469i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.939iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 + 3.31iT - 29T^{2} \) |
| 31 | \( 1 - 7.57iT - 31T^{2} \) |
| 37 | \( 1 - 8.21iT - 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 + 1.08iT - 43T^{2} \) |
| 47 | \( 1 + 8.51iT - 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 2.37iT - 67T^{2} \) |
| 71 | \( 1 + 8.51iT - 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.804694875246104726231320713548, −8.702318322446078895701817681486, −8.190346568785188190003261737471, −7.00619049468367205723168427453, −6.25187237585274430686140091309, −5.79705257062935563937592921791, −4.81778670525470598510384533450, −3.50403203086713283618017514099, −1.92288101889555719365064140267, −1.18277084709989105097977009393,
0.65134312196438047454452807659, 1.94642013665966154864630933356, 3.14381546918034367106921579597, 4.62077441384686236998510938631, 5.40122985469402705170135497209, 6.23670175329624005844093474195, 6.92324010523931214145585565778, 7.62683864292916587890118316334, 9.071803848999005552169750811500, 9.393088376839329309504903071703