L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1 − 2i)5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.707 + 2.12i)10-s − 1.96·11-s + (−0.707 + 0.707i)12-s + (2.37 + 2.37i)13-s + (0.707 − 2.12i)15-s − 1.00·16-s + (−3.38 + 3.38i)17-s + (0.707 − 0.707i)18-s + 3.41·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.447 − 0.894i)5-s − 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.223 + 0.670i)10-s − 0.592·11-s + (−0.204 + 0.204i)12-s + (0.659 + 0.659i)13-s + (0.182 − 0.547i)15-s − 0.250·16-s + (−0.821 + 0.821i)17-s + (0.166 − 0.166i)18-s + 0.783·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.047799784\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047799784\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + (-2.37 - 2.37i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.38 - 3.38i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.22iT - 29T^{2} \) |
| 31 | \( 1 - 1.98iT - 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-7.81 + 7.81i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.93 - 4.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.39 + 8.39i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 - 3.92iT - 61T^{2} \) |
| 67 | \( 1 + (-2.36 - 2.36i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 + (2.01 + 2.01i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.02T + 89T^{2} \) |
| 97 | \( 1 + (5.08 - 5.08i)T - 97iT^{2} \) |
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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
−9.572410489596361516702541765441, −8.836641622920839526160304592113, −8.204906411762903760107791051026, −7.63811818249353316003470500550, −6.40111414957789815463729119910, −5.27489411001228893581725991540, −4.28591000365598626789938305462, −3.68830941287985854200574348595, −2.41700934935245982354471016542, −1.23097487908483310228645682340,
0.51430020925691631995300971195, 2.24533212342993507181855613445, 3.12059184006213621258750984585, 4.25385373404590702580405392364, 5.53564862795786289990259751556, 6.31720926979553833649614650759, 7.26389995562047485385046421771, 7.61729281976670070287227768623, 8.484030821967430812433850727600, 9.212788384947990817928066044841