L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1.19 − 1.89i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.493 − 2.18i)10-s − 1.97·11-s + (−0.707 + 0.707i)12-s + (−2.19 − 2.19i)13-s + (0.493 − 2.18i)15-s − 1.00·16-s + (−3.25 + 3.25i)17-s + (−0.707 + 0.707i)18-s − 4.21·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.533 − 0.845i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.156 − 0.689i)10-s − 0.596·11-s + (−0.204 + 0.204i)12-s + (−0.608 − 0.608i)13-s + (0.127 − 0.563i)15-s − 0.250·16-s + (−0.789 + 0.789i)17-s + (−0.166 + 0.166i)18-s − 0.967·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5726249571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5726249571\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.19 + 1.89i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.97T + 11T^{2} \) |
| 13 | \( 1 + (2.19 + 2.19i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.25 - 3.25i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + (4.15 - 4.15i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (1.96 + 1.96i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.55iT - 41T^{2} \) |
| 43 | \( 1 + (6.33 - 6.33i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.29 + 4.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.08 + 8.08i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 + (-3.89 - 3.89i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 + (10.7 + 10.7i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.55iT - 79T^{2} \) |
| 83 | \( 1 + (-9.52 - 9.52i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 + (1.48 - 1.48i)T - 97iT^{2} \) |
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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
−9.848957911434826485088817103305, −8.867338681196486247025096945187, −8.282733647208186762025514956500, −7.70581749359504056531567430940, −6.72180245951268229481338914913, −5.58339973399417929857694686588, −4.92341574648798596681671503335, −4.12265718178334168312225667111, −3.28963088559298613670188566191, −1.97503808825808439975101503220,
0.16435747898454862091969580807, 2.25699491352490323749389898688, 2.59734771296872833399298795897, 3.93024993443090987342597246561, 4.52521819658996519063387489361, 5.83356477235334087356049074480, 6.73207537652288870012685960094, 7.30477793823545914689194375924, 8.260700020144303140352589016671, 9.054202150436506062872470364920