L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (2.22 + 0.243i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (1.39 + 1.74i)10-s − 5.33·11-s + (0.707 − 0.707i)12-s + (3.82 + 3.82i)13-s + (−1.39 − 1.74i)15-s − 1.00·16-s + (−5.19 + 5.19i)17-s + (−0.707 + 0.707i)18-s − 3.04·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.994 + 0.108i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.442 + 0.551i)10-s − 1.60·11-s + (0.204 − 0.204i)12-s + (1.06 + 1.06i)13-s + (−0.361 − 0.450i)15-s − 0.250·16-s + (−1.26 + 1.26i)17-s + (−0.166 + 0.166i)18-s − 0.697·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579057195\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579057195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.22 - 0.243i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 + (-3.82 - 3.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.19 - 5.19i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + (1.63 - 1.63i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 - 6.37iT - 31T^{2} \) |
| 37 | \( 1 + (-2.11 - 2.11i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + (-1.27 + 1.27i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.75 + 8.75i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.91 - 4.91i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 + 0.777iT - 61T^{2} \) |
| 67 | \( 1 + (5.72 + 5.72i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 + (-5.03 - 5.03i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 + (-2.41 - 2.41i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.19T + 89T^{2} \) |
| 97 | \( 1 + (4.72 - 4.72i)T - 97iT^{2} \) |
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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
−9.836378503385035411372584865169, −8.691233948936327447990467252341, −8.229380418970045819144843569100, −7.08135285419764490166938248518, −6.31579420213726897768404260313, −5.91390397552145967197382808136, −4.93767161590473250960244530775, −4.05393178043500898610192175641, −2.60715483638275346859170034139, −1.73825733447043262473214833592,
0.52091996337644473069490293193, 2.21316676833868331495506079249, 2.93286453261007002752643181696, 4.24462424188308330777727986208, 5.13310730213984838891706183087, 5.69043032520078784842232236362, 6.42567713335701224869213380920, 7.57910152118073833017466473743, 8.710407030530354534553125624942, 9.344030072870565737077689345953